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...hods to obtain upper bounds on ASq (n, d) are various types of linear programming methods. The linear programming method suggested in [50] is different from the classic linear programming of Delsarte =-=[36]-=-. The classic linear programming does not yield any improvements, when applied for bounds on codes in Gq(n, k) or in Pq(n). A somehow stronger method is the semidefinite programming [90, 91] as sugges...

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...ance or the injection distance. Quite naturally also list-decoding algorithms were developed for Grassmannian codes (constructed either by linearlized polynomials or as lifted rank-metric codes), e.g =-=[2, 59, 60, 73, 74, 75, 76, 85, 111]-=-. Also, this direction of research has many problems for future research. Research problem 56. Suggest new classes of large Grassmannian codes, subspace codes (with the subspace distance or the inject...

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...s. In recent years there has been an increasing interest in subspace codes as a result of their application to error-correction in random network coding as was demonstrated by Koetter and Kschischang =-=[70]-=-. But, the interest in these codes has been also before this application, since Grassmannian codes are q-analogs of the well studied constant weight codes [23]. For example, the nonexistence of nontri...

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...matrices over Fq; they form a linear subspace with dimension ̺ of Fk×ℓq , and for each two distinct codewords A and B we have that dR(A,B) ≥ δ. For a [k × ℓ, ̺, δ] rank-metric code C it was proved in =-=[38, 54, 86]-=- that ̺ ≤ min{k(ℓ− δ + 1), ℓ(k − δ + 1)} . (1) This bound, called the Singleton bound for rank-metric codes, is attained for all feasible parameters. The codes which attain this bound are called maxim...

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...q . As for lower bounds on Aq(n, δ, k), in [70] there is a construction, of codes based on linearlized polynomials, which yields the bound Aq(n, δ, k) ≥ q(n−k)(k−δ+1). The same bound was developed in =-=[98]-=- by using lifted rank-metric codes. In this context we define the rank distance and rank-metric codes which play an important role in the discussion on subspace codes. For two k × ℓ matrices A and B o...

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...and the Grassmannian. We will consider three topics: Gray codes, self-complements codes, and linear codes. 14.1 Gray Codes Gray codes have many applications and they are defined on variety of objects =-=[88]-=-. A Gray code in Pq(n) or Gq(n, k) is a path in the related graphs G(PSq (n)) and G(Gq(n, k)), respectively. In G(PSq (n)) the vertices represent the subspaces of F n q . Two vertices X and Y are conn...

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... demonstrated by Koetter and Kschischang [70]. But, the interest in these codes has been also before this application, since Grassmannian codes are q-analogs of the well studied constant weight codes =-=[23]-=-. For example, the nonexistence of nontrivial perfect codes in the Grassmann scheme was proved in [27, 79]. The well-known concept of q-analogs replaces subsets by subspaces of a vector space over a f...

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...matrices over Fq; they form a linear subspace with dimension ̺ of Fk×ℓq , and for each two distinct codewords A and B we have that dR(A,B) ≥ δ. For a [k × ℓ, ̺, δ] rank-metric code C it was proved in =-=[38, 54, 86]-=- that ̺ ≤ min{k(ℓ− δ + 1), ℓ(k − δ + 1)} . (1) This bound, called the Singleton bound for rank-metric codes, is attained for all feasible parameters. The codes which attain this bound are called maxim...

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...matrices over Fq; they form a linear subspace with dimension ̺ of Fk×ℓq , and for each two distinct codewords A and B we have that dR(A,B) ≥ δ. For a [k × ℓ, ̺, δ] rank-metric code C it was proved in =-=[38, 54, 86]-=- that ̺ ≤ min{k(ℓ− δ + 1), ℓ(k − δ + 1)} . (1) This bound, called the Singleton bound for rank-metric codes, is attained for all feasible parameters. The codes which attain this bound are called maxim...

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...g of Delsarte [36]. The classic linear programming does not yield any improvements, when applied for bounds on codes in Gq(n, k) or in Pq(n). A somehow stronger method is the semidefinite programming =-=[90, 91]-=- as suggested in [5] for the projective space. This method and the one in [50] yield all the best known upper bounds on ASq (n, d), when in most cases the semidefinite programming yields better bounds...

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...g of Delsarte [36]. The classic linear programming does not yield any improvements, when applied for bounds on codes in Gq(n, k) or in Pq(n). A somehow stronger method is the semidefinite programming =-=[90, 91]-=- as suggested in [5] for the projective space. This method and the one in [50] yield all the best known upper bounds on ASq (n, d), when in most cases the semidefinite programming yields better bounds...

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... dimension code C. If C is an MRD code then C is called a 4 lifted MRD code [98]. This code will be denoted by CMRD. This code is not maximal and it can be extended by using a multilevel construction =-=[48]-=- as described in the next section. An upper bound on the size of a code which contains CMRD can be found in [49]. Codes based on linearlized polynomials, where each code contains the related code base...

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...mension. An (n, δ, k)q code is a subset of Gq(n, k) with minimum Grassmannian distance δ. Let Aq(n, δ, k) denote the maximum size of an (n, δ, k)q code. Koetter and Kschischang [70], Etzion and Vardy =-=[50]-=- developed several upper bounds on Aq(n, δ, k). For a subspace code C we define the orthogonal complement C⊥ as the code which consists of the dual subspaces of C, i.e. C⊥ def ={X⊥ : X ∈ C}. C and C⊥ ...

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...k) δ k3 2 3 a9a 2 b77− 81c a21a Bounds on A2(7, δ, k) δ k3 2 3 d17d 2 e329− 381f g31h 6 Bounds on A2(8, δ, k) δ k4 3 2 4 a17a 3 i257− 289j d34d 2 k4797− 6477f e1312− 1493c a85a • a - Theorem 8. • b - =-=[71]-=-. • c - [50]. • d - [41]. • e - [22]. • f - Theorem 2. • g - Theorem 10. • h - Theorem 9. • i - The Multilevel Construction with CMRD. • j - Theorem 1. • k - [49]. Some of the specific values of Aq(n,...

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...the case of error-correcting codes in the projective space (which are q-packing designs) there are some basic bounds on the size of a q-covering design. The first one is the q-analog Schönheim bound =-=[92]-=- which was given in [52]. Theorem 17. Cq(n, k, r) ≥ ⌈ qn−1 qk−1 Cq(n− 1, k − 1, r − 1) ⌉ . The basic covering bound is given in the following theorem [52]. Theorem 18. Cq(n, k, r) ≥ [nr]q [kr]q with e...

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...stems were subject to an extensive research in combinatorial designs [33]. A Steiner system is also equivalent to an optimal constant weight codes in the Hamming scheme. Cameron [25, 26] and Delsarte =-=[37]-=- have extended the notions of block design and Steiner systems to vector spaces. A Steiner structure (q-Steiner system) Sq(t, k, n) is a collection S of elements from Gq(n, k) (called blocks) such tha...

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...tric An (n, d)Iq code is a subspace code in Pq(n) with minimum injection distance d. The injection distance is the one which is more useful, than the subspace distance, from a practical point of view =-=[99]-=-. Let AIq(n, d) denote the maximum number of codewords in an (n, d) I q code. The injection distance is the q-analog of the asymmetric distance [68]. Also, the related graph G(PIq (n)) is not distance...

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...s of a very special interest. This value has a special interest since (n, k, k)q codes have applications as byte-correcting codes [43, 65]. Decoding of such constant dimension codes was considered in =-=[58, 77]-=- The known upper and lower bounds on Aq(n, k, k) are summarized in the following theorems. The first three well-known theorems can be found in [50]. Theorem 8. If k divides n then Aq(n, k, k) = qn−1 q...

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...code which contains CMRD can be found in [49]. Codes based on linearlized polynomials, where each code contains the related code based on linearlized polynomial constructed in [70], were developed in =-=[100]-=-. But, these codes are smaller in size then the codes obtained by the multilevel construction. Another family of Grassmannian codes are codes which admit a certain automorphism group. These kind of co...

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...the existence question of perfect subspace codes with the subspace distance. It was proved in [50] that such codes do not exist. The Johnson space and the Grassmann space admit diameter-perfect codes =-=[4]-=-. All such diameter-perfect codes are optimal for their parameters. Unfortunately, the definition of diameter-perfect codes does not extend to the projective space Pq(n), since the size of a sphere in...

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... 1 ≤ k ≤ n, then Cq(n, k, 1) = ⌈ qn−1 qk−1 ⌉ . Theorem 22. If 1 ≤ r ≤ n− 1, then Cq(n, n− 1, r) = qr+1−1 q−1 . Theorem 22 was proved before in the context of projective geometry by Bose and Burton in =-=[16]-=-. Another lower bound was given in [40] by considering sets of lines in PG(2s, q) contained in s-subspaces. Theorem 23. Cq(2s+ 1, 2s− 1, s) ≥ q2s+2−q2 q2−1 + q s+1−1 q−1 for every integer s ≥ 2. Metsc...

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...q) are known for many years. For q = 2 and odd n there is an 1parallelism in PG(n, 2). Such a parallelism was found in the context of Preparata codes and it is known that many such parallelisms exist =-=[6, 7, 119]-=-. For any other power of a prime q, if n = 2i − 1, i ≥ 2, then an 1-parallelism was shown in [8]. In the last forty years no new parameters for 1-parallelisms were shown until recently, when an 1-para...

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...relation for subspace codes. However, a specific type of rank-metric codes, namely constant rank codes have an important direct application to constant dimension codes. These codes were considered in =-=[56]-=-. A constant rank code is a rank-metric code in which all codewords have the same rank. Let ARq (m,n, d, r) be the maximum number of codewords in a constant rank code with constant rank r and minimum ...

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...tive geometry. A set T of t-subspaces in PG(n, q) such that each s-subspace contains at least one element of T is called 22 a blocking set. Such a design is a q-analog of the well-known Turán design =-=[34, 35]-=-. The dual subspaces of the subspaces in a blocking set form a q-covering design Cq(n+1, n− t, n− s). Blocking sets were considered for example in [82, 83]. We note that blocking sets have also some d...

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...ant dimension code which attains the bound of Aq(n, k − t + 1, k). Similarly, to Steiner systems, simple necessary divisibility conditions for the existence of a given Steiner structure are developed =-=[93]-=-. Theorem 6. If a Steiner structure Sq(t, k, n) exists then for each i, 1 ≤ i ≤ t− 1, a Steiner structure Sq(t− i, k − i, n− i) exists. Corollary 1. If a Steiner structure Sq(t, k, n) exists then for ...

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...(n, k) such that each t-dimensional subspace of Gq(n, t) is contained in exactly λ blocks of B. If B contains all the k-dimensional subspaces of Gq(n, k) then the design is said to be trivial. Thomas =-=[106]-=- was the first to find nontrivial t-design over Fq, which are not spreads. The work of Thomas has motivated other research work to explore this topic and more t-designs over Fq were found [20, 67, 84,...

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...pace. q-analogs of various combinatorial objects are well known [115, pp. 325-332]. Of special interest are q-analogs in extremal combinatoric as well as other well-known combinatorial problems, e.g. =-=[14, 15, 28, 29, 53, 66]-=-. The related techniques and results might be of usage in coding theory. It turns out that the natural measure of distance in Pq(n) is given by dS(X, Y ) def = dimX + dimY − 2 dim ( X ∩Y ) , for all X...

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... combinatorial structure as the other two graphs. Another interesting problem in this context is the q-analog of the middle levels problem which is a well-known unsolved problem for the Hamming graph =-=[89]-=-. The q-analog problems are presented as follows. Research problem 96. Is there a cycle in G(PSq (2k+1)) which contains all the k-dimensional subspaces and all the (k + 1)-dimensional subspaces? Resea...

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...qn − qk(qr − 1)− 1 qk − 1 We note that one method to obtain the lower bound of Theorem 10 is to apply the Multilevel Construction. The next theorem was proved in [65] for q = 2 and for any other q in =-=[9]-=-. 18 Theorem 11. If n ≡ 1 (mod k) then Aq(n, k, k) = qn−q qk−1 − q + 1 = ∑n−1 k −1 i=1 q ik+1 + 1. The bound of Theorem 11 is attained with CMRD to which one subspace is added. By Theorems 8 and 11, t...

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...his application, since Grassmannian codes are q-analogs of the well studied constant weight codes [23]. For example, the nonexistence of nontrivial perfect codes in the Grassmann scheme was proved in =-=[27, 79]-=-. The well-known concept of q-analogs replaces subsets by subspaces of a vector space over a finite field and their orders by the dimensions of the subspaces. In particular, the q-analog of a constant...

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...q) are known for many years. For q = 2 and odd n there is an 1parallelism in PG(n, 2). Such a parallelism was found in the context of Preparata codes and it is known that many such parallelisms exist =-=[6, 7, 119]-=-. For any other power of a prime q, if n = 2i − 1, i ≥ 2, then an 1-parallelism was shown in [8]. In the last forty years no new parameters for 1-parallelisms were shown until recently, when an 1-para...

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...m was found in the context of Preparata codes and it is known that many such parallelisms exist [6, 7, 119]. For any other power of a prime q, if n = 2i − 1, i ≥ 2, then an 1-parallelism was shown in =-=[8]-=-. In the last forty years no new parameters for 1-parallelisms were shown until recently, when an 1-parallelism in PG(5,3) was proved to exist in [51]. A k-parallelism for k > 1 was not known until a ...

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...vides n is a Steiner structure Sq(1, k, n). The value of Aq(n, k, k) is of a very special interest. This value has a special interest since (n, k, k)q codes have applications as byte-correcting codes =-=[43, 65]-=-. Decoding of such constant dimension codes was considered in [58, 77] The known upper and lower bounds on Aq(n, k, k) are summarized in the following theorems. The first three well-known theorems can...

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...her measure of distance, called the injection distance given by dI(X, Y ) = max{dim(X), dim(Y )} − dim(X ∩ Y ) . The injection distance is the q-analog of the asymmetric distance between binary words =-=[42, 47, 94]-=-. Both, the subspace distance and the injection distance are metrics. When X and Y have the same dimension k, the subspace metric and the injection metric coincide. If X, Y ∈ Gq(n, k) then we can defi...

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.... This will be further demonstrated and explained at the end of this section. Also, as in the case of the subspace distance, classic lower bounds such as the Gilbert-Varshamov bound were developed in =-=[55, 68]-=-. As for upper bounds, a linear programming was developed in [3], and in [5] there is a modification of the linear programming given in [50]. A semidefinite programming bound was also given in [5] and...

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...ance or the injection distance. Quite naturally also list-decoding algorithms were developed for Grassmannian codes (constructed either by linearlized polynomials or as lifted rank-metric codes), e.g =-=[2, 59, 60, 73, 74, 75, 76, 85, 111]-=-. Also, this direction of research has many problems for future research. Research problem 56. Suggest new classes of large Grassmannian codes, subspace codes (with the subspace distance or the inject...

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... that the largest subset of Pq(n) on which a complement can be defined is the set Vq(n) = {X ∈ Pq(n) : X ∩X⊥ = {0}}; or disprove this claim. A closed-form expression for |Vq(n)| was given by Sendrier =-=[103]-=-. Using the results of [103], it can be shown that the size of Vq(n) is proportional to |Pq(n)|, specifically: lim n→∞ |Vq(n)| |Pq(n)| = ∞∏ i=1 1 1 + q−i . The limit converge to 0.4194 . . . when q = ...

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...pace. q-analogs of various combinatorial objects are well known [115, pp. 325-332]. Of special interest are q-analogs in extremal combinatoric as well as other well-known combinatorial problems, e.g. =-=[14, 15, 28, 29, 53, 66]-=-. The related techniques and results might be of usage in coding theory. It turns out that the natural measure of distance in Pq(n) is given by dS(X, Y ) def = dimX + dimY − 2 dim ( X ∩Y ) , for all X...

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...her measure of distance, called the injection distance given by dI(X, Y ) = max{dim(X), dim(Y )} − dim(X ∩ Y ) . The injection distance is the q-analog of the asymmetric distance between binary words =-=[42, 47, 94]-=-. Both, the subspace distance and the injection distance are metrics. When X and Y have the same dimension k, the subspace metric and the injection metric coincide. If X, Y ∈ Gq(n, k) then we can defi...

Citation Context

...vides n is a Steiner structure Sq(1, k, n). The value of Aq(n, k, k) is of a very special interest. This value has a special interest since (n, k, k)q codes have applications as byte-correcting codes =-=[43, 65]-=-. Decoding of such constant dimension codes was considered in [58, 77] The known upper and lower bounds on Aq(n, k, k) are summarized in the following theorems. The first three well-known theorems can...

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...lassic linear programming does not yield any improvements, when applied for bounds on codes in Gq(n, k) or in Pq(n). A somehow stronger method is the semidefinite programming [90, 91] as suggested in =-=[5]-=- for the projective space. This method and the one in [50] yield all the best known upper bounds on ASq (n, d), when in most cases the semidefinite programming yields better bounds. The following list...

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...ement of S. Steiner systems were subject to an extensive research in combinatorial designs [33]. A Steiner system is also equivalent to an optimal constant weight codes in the Hamming scheme. Cameron =-=[25, 26]-=- and Delsarte [37] have extended the notions of block design and Steiner systems to vector spaces. A Steiner structure (q-Steiner system) Sq(t, k, n) is a collection S of elements from Gq(n, k) (calle...

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...ance or the injection distance. Quite naturally also list-decoding algorithms were developed for Grassmannian codes (constructed either by linearlized polynomials or as lifted rank-metric codes), e.g =-=[2, 59, 60, 73, 74, 75, 76, 85, 111]-=-. Also, this direction of research has many problems for future research. Research problem 56. Suggest new classes of large Grassmannian codes, subspace codes (with the subspace distance or the inject...

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...his application, since Grassmannian codes are q-analogs of the well studied constant weight codes [23]. For example, the nonexistence of nontrivial perfect codes in the Grassmann scheme was proved in =-=[27, 79]-=-. The well-known concept of q-analogs replaces subsets by subspaces of a vector space over a finite field and their orders by the dimensions of the subspaces. In particular, the q-analog of a constant...

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...1 was extended for the case where q = 2 and k = 3 in [41] as follows. Theorem 12. If n ≡ c (mod 3) then A2(n, 3, 3) = 2n−2c 7 − c. The upper bound implied by Theorem 11 was improved for some cases in =-=[39]-=- in which a transformation, of partial spreads into orthogonal arrays of strength two, is considered. Theorem 13. If n = kℓ + c with 0 < c < k, then Aq(n, k, k) ≤ ∑ℓ−1 i=0 q ik+c − Ω − 1, where 2Ω = √...

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...ivated other research work to explore this topic and more t-designs over Fq were found [20, 67, 84, 104, 105, 107]. Another type of design over Fq which was defined is the subspace transversal design =-=[49]-=-. It is not a direct q-analog of a transversal design as will be explained in the sequel. Let V(n,k) be the set of nonzero vectors of Fnq whose first k entries form a nonzero vector. For a given X ∈ G...

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...ement of S. Steiner systems were subject to an extensive research in combinatorial designs [33]. A Steiner system is also equivalent to an optimal constant weight codes in the Hamming scheme. Cameron =-=[25, 26]-=- and Delsarte [37] have extended the notions of block design and Steiner systems to vector spaces. A Steiner structure (q-Steiner system) Sq(t, k, n) is a collection S of elements from Gq(n, k) (calle...

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...i) exists. Corollary 1. If a Steiner structure Sq(t, k, n) exists then for all 0 ≤ i ≤ t− 1. [ n−i t−i ] q[ k−i t−i ] q must be integers. Steiner structures and Steiner systems are highly related. In =-=[52, 93]-=- there are some constructions of Steiner systems derived from Steiner structures. Further research on Steiner structures seems to be fascinating, but also extremely difficult. We list some interesting...

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...ding is trivial by using the encoding of the related rank-metric code. If the code is constructed by the Multilevel Construction then the encoding is slightly more complicated and it was described in =-=[69]-=-. Finally, encoding of all the Grassmannian space was described in [80, 95]. Research problem 60. Find better encoding algorithms for subspace codes, the Grassmannian Gq(n, k), or the projective space...

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...exist if and only if k divides n. These are called spreads and they will be discussed in the next section. Thomas [107] showed that certain kind of Steiner structures S2(2, 3, 7) cannot exist. Metsch =-=[81]-=- conjectured that nontrivial Steiner structures with t ≥ 2 do not exist. Steiner structures appear also in connection of diameter perfect codes in the Grassmann scheme. It was proved in [4] that the o...

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...If the code is constructed by the Multilevel Construction then the encoding is slightly more complicated and it was described in [69]. Finally, encoding of all the Grassmannian space was described in =-=[80, 95]-=-. Research problem 60. Find better encoding algorithms for subspace codes, the Grassmannian Gq(n, k), or the projective space Pq(n). 10 Designs over GF(q) q-analog of Steiner systems are one type of q...

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...ven by construction, on the size of a q-covering design was proved in [52]. Theorem 25. Cq(n, k, r) ≤ qn−kCq(n− 1, k − 1, r − 1) + Cq(n− 1, k, r). Normal spreads [72], also known as geometric spreads =-=[12]-=-, are used to prove the following values of Cq(n, k, r) [13]. Theorem 26. Cq(vm+ δ, vm−m+ δ, v − 1) = qvm−1 qm−1 for all v ≥ 2, m ≥ 2, and δ ≥ 0. The next theorem given in [52] is used infinitely many...

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...pace. q-analogs of various combinatorial objects are well known [115, pp. 325-332]. Of special interest are q-analogs in extremal combinatoric as well as other well-known combinatorial problems, e.g. =-=[14, 15, 28, 29, 53, 66]-=-. The related techniques and results might be of usage in coding theory. It turns out that the natural measure of distance in Pq(n) is given by dS(X, Y ) def = dimX + dimY − 2 dim ( X ∩Y ) , for all X...

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...y codes are used as a tool in various aspects of coding theory. The q-analog was considered in [19]. Various related problems concerning complements of subspaces over Fq were considered before, e. g. =-=[30, 31, 32]-=-. Definition 1. Let U be a subset of Pq(n) and let Uk := U ∩Gq(n, k). We say that a function f : U → U is a complement on U (and denote X = f(X) for all X ∈ U) if f has the following properties: P1. X...

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... mainly since the value of Aq(n, δ, 2) is known for all parameters. Another family of codes which was considered, even so the codes were not as large as in previous constructions, are the orbit codes =-=[78, 109, 110, 114]-=-. This family of codes might deserve further attention in the future. Another line of research for Grassmannain codes is based on Schubert calculus and Plücker coordinates. These were considered for ...

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...ker coordinates. These were considered for example in [57, 111, 113] and their further research might lead to new interesting results. Lexicodes in the Grassmannian and their search were discussed in =-=[96]-=-. Codes which are able to correct also errors in coordinates (such as deletion or localized errors) are considered in [24, 101]. A question on the size of equidistant Grassmannian codes was asked in [...

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...Thomas [106] was the first to find nontrivial t-design over Fq, which are not spreads. The work of Thomas has motivated other research work to explore this topic and more t-designs over Fq were found =-=[20, 67, 84, 104, 105, 107]-=-. Another type of design over Fq which was defined is the subspace transversal design [49]. It is not a direct q-analog of a transversal design as will be explained in the sequel. Let V(n,k) be the se...

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...Thomas [106] was the first to find nontrivial t-design over Fq, which are not spreads. The work of Thomas has motivated other research work to explore this topic and more t-designs over Fq were found =-=[20, 67, 84, 104, 105, 107]-=-. Another type of design over Fq which was defined is the subspace transversal design [49]. It is not a direct q-analog of a transversal design as will be explained in the sequel. Let V(n,k) be the se...

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...Thomas [106] was the first to find nontrivial t-design over Fq, which are not spreads. The work of Thomas has motivated other research work to explore this topic and more t-designs over Fq were found =-=[20, 67, 84, 104, 105, 107]-=-. Another type of design over Fq which was defined is the subspace transversal design [49]. It is not a direct q-analog of a transversal design as will be explained in the sequel. Let V(n,k) be the se...

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...y codes are used as a tool in various aspects of coding theory. The q-analog was considered in [19]. Various related problems concerning complements of subspaces over Fq were considered before, e. g. =-=[30, 31, 32]-=-. Definition 1. Let U be a subset of Pq(n) and let Uk := U ∩Gq(n, k). We say that a function f : U → U is a complement on U (and denote X = f(X) for all X ∈ U) if f has the following properties: P1. X...

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...Thomas [106] was the first to find nontrivial t-design over Fq, which are not spreads. The work of Thomas has motivated other research work to explore this topic and more t-designs over Fq were found =-=[20, 67, 84, 104, 105, 107]-=-. Another type of design over Fq which was defined is the subspace transversal design [49]. It is not a direct q-analog of a transversal design as will be explained in the sequel. Let V(n,k) be the se...

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...n [82]. The most basic upper bound, given by construction, on the size of a q-covering design was proved in [52]. Theorem 25. Cq(n, k, r) ≤ qn−kCq(n− 1, k − 1, r − 1) + Cq(n− 1, k, r). Normal spreads =-=[72]-=-, also known as geometric spreads [12], are used to prove the following values of Cq(n, k, r) [13]. Theorem 26. Cq(vm+ δ, vm−m+ δ, v − 1) = qvm−1 qm−1 for all v ≥ 2, m ≥ 2, and δ ≥ 0. The next theorem...

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...s a q-analog of the well-known Turán design [34, 35]. The dual subspaces of the subspaces in a blocking set form a q-covering design Cq(n+1, n− t, n− s). Blocking sets were considered for example in =-=[82, 83]-=-. We note that blocking sets have also some different definitions (and maybe more popular definitions which define other structures which are not q-coverings). Similarly, to the case of error-correcti...

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...ance or the injection distance. Quite naturally also list-decoding algorithms were developed for Grassmannian codes (constructed either by linearlized polynomials or as lifted rank-metric codes), e.g =-=[2, 59, 60, 73, 74, 75, 76, 85, 111]-=-. Also, this direction of research has many problems for future research. Research problem 56. Suggest new classes of large Grassmannian codes, subspace codes (with the subspace distance or the inject...

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...ters for 1-parallelisms were shown until recently, when an 1-parallelism in PG(5,3) was proved to exist in [51]. A k-parallelism for k > 1 was not known until a 2-parallelism in PG(5, 2) was shown in =-=[87]-=-. Clearly, such parallelisms can be described in terms of spreads in the Grassmannian. As it seems to be extremely difficult, we consider two problems which are generalizations of the parallelism prob...

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...her measure of distance, called the injection distance given by dI(X, Y ) = max{dim(X), dim(Y )} − dim(X ∩ Y ) . The injection distance is the q-analog of the asymmetric distance between binary words =-=[42, 47, 94]-=-. Both, the subspace distance and the injection distance are metrics. When X and Y have the same dimension k, the subspace metric and the injection metric coincide. If X, Y ∈ Gq(n, k) then we can defi...

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...Thomas [106] was the first to find nontrivial t-design over Fq, which are not spreads. The work of Thomas has motivated other research work to explore this topic and more t-designs over Fq were found =-=[20, 67, 84, 104, 105, 107]-=-. Another type of design over Fq which was defined is the subspace transversal design [49]. It is not a direct q-analog of a transversal design as will be explained in the sequel. Let V(n,k) be the se...

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...section. Also, as in the case of the subspace distance, classic lower bounds such as the Gilbert-Varshamov bound were developed in [55, 68]. As for upper bounds, a linear programming was developed in =-=[3]-=-, and in [5] there is a modification of the linear programming given in [50]. A semidefinite programming bound was also given in [5] and it was shown that the bounds obtained are very similar to those...

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...proved in [52]. Theorem 25. Cq(n, k, r) ≤ qn−kCq(n− 1, k − 1, r − 1) + Cq(n− 1, k, r). Normal spreads [72], also known as geometric spreads [12], are used to prove the following values of Cq(n, k, r) =-=[13]-=-. Theorem 26. Cq(vm+ δ, vm−m+ δ, v − 1) = qvm−1 qm−1 for all v ≥ 2, m ≥ 2, and δ ≥ 0. The next theorem given in [52] is used infinitely many times once an exact bound for some given parameters is know...

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...ll X ∈ U . P4. dS(X, Y ) = dS(X, Y ) for all X, Y ∈ U . The existence problems of complements in Pq(n) was considered in [19]. Some of the results are based on representation of subspaces by lattices =-=[17]-=-. The main open problem which remains unsolved in this discussion is our next open problem. Research problem 98. Prove that the largest subset of Pq(n) on which a complement can be defined is the set ...

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...morphisms of constant dimension codes were studied in [108]. Constructions for small dimensions might be attractive in this context. Interesting codes admitting some automorphisms were constructed in =-=[22]-=-. Some of these codes have an interesting combinatorial structure and some were found only by computer search. These were used to obtain lower bounds on A2(n, 2, 3). Lower bounds on Aq(n, 2, 3) were a...

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...tive geometry. A set T of t-subspaces in PG(n, q) such that each s-subspace contains at least one element of T is called 22 a blocking set. Such a design is a q-analog of the well-known Turán design =-=[34, 35]-=-. The dual subspaces of the subspaces in a blocking set form a q-covering design Cq(n+1, n− t, n− s). Blocking sets were considered for example in [82, 83]. We note that blocking sets have also some d...

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... 2i − 1, i ≥ 2, then an 1-parallelism was shown in [8]. In the last forty years no new parameters for 1-parallelisms were shown until recently, when an 1-parallelism in PG(5,3) was proved to exist in =-=[51]-=-. A k-parallelism for k > 1 was not known until a 2-parallelism in PG(5, 2) was shown in [87]. Clearly, such parallelisms can be described in terms of spreads in the Grassmannian. As it seems to be ex...

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...ance or the injection distance. Quite naturally also list-decoding algorithms were developed for Grassmannian codes (constructed either by linearlized polynomials or as lifted rank-metric codes), e.g =-=[2, 59, 60, 73, 74, 75, 76, 85, 111]-=-. Also, this direction of research has many problems for future research. Research problem 56. Suggest new classes of large Grassmannian codes, subspace codes (with the subspace distance or the inject...

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...t two sections. A comprehensive description and discussion on the Multilevel Construction can be found in [48]. Some improvements which enable to add more codewords for the final code can be found in =-=[97, 112]-=-. Research problem 13. Find a method which combine the Multilevel Construction with more concepts to obtain larger codes. Research problem 14. What is the best way to choose a skeleton code for the Mu...

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...pace. q-analogs of various combinatorial objects are well known [115, pp. 325-332]. Of special interest are q-analogs in extremal combinatoric as well as other well-known combinatorial problems, e.g. =-=[14, 15, 28, 29, 53, 66]-=-. The related techniques and results might be of usage in coding theory. It turns out that the natural measure of distance in Pq(n) is given by dS(X, Y ) def = dimX + dimY − 2 dim ( X ∩Y ) , for all X...

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...pace. q-analogs of various combinatorial objects are well known [115, pp. 325-332]. Of special interest are q-analogs in extremal combinatoric as well as other well-known combinatorial problems, e.g. =-=[14, 15, 28, 29, 53, 66]-=-. The related techniques and results might be of usage in coding theory. It turns out that the natural measure of distance in Pq(n) is given by dS(X, Y ) def = dimX + dimY − 2 dim ( X ∩Y ) , for all X...

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...k−1 ⌉ . Theorem 22. If 1 ≤ r ≤ n− 1, then Cq(n, n− 1, r) = qr+1−1 q−1 . Theorem 22 was proved before in the context of projective geometry by Bose and Burton in [16]. Another lower bound was given in =-=[40]-=- by considering sets of lines in PG(2s, q) contained in s-subspaces. Theorem 23. Cq(2s+ 1, 2s− 1, s) ≥ q2s+2−q2 q2−1 + q s+1−1 q−1 for every integer s ≥ 2. Metsch [82] also gave a construction for a s...

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...1c a21a Bounds on A2(7, δ, k) δ k3 2 3 d17d 2 e329− 381f g31h 6 Bounds on A2(8, δ, k) δ k4 3 2 4 a17a 3 i257− 289j d34d 2 k4797− 6477f e1312− 1493c a85a • a - Theorem 8. • b - [71]. • c - [50]. • d - =-=[41]-=-. • e - [22]. • f - Theorem 2. • g - Theorem 10. • h - Theorem 9. • i - The Multilevel Construction with CMRD. • j - Theorem 1. • k - [49]. Some of the specific values of Aq(n, δ, k) can be of special...

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...e subspace distance, from a practical point of view [99]. Let AIq(n, d) denote the maximum number of codewords in an (n, d) I q code. The injection distance is the q-analog of the asymmetric distance =-=[68]-=-. Also, the related graph G(PIq (n)) is not distance regular which makes the analysis of some bounds more difficult as in the case of the subspace distance and the related graph G(PSq (n)). Similarly ...

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...ing results. Lexicodes in the Grassmannian and their search were discussed in [96]. Codes which are able to correct also errors in coordinates (such as deletion or localized errors) are considered in =-=[24, 101]-=-. A question on the size of equidistant Grassmannian codes was asked in [50]. In an equidistant code, any two codewords have the same distance, which is clearly the minimum distance of the code. This ...

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...ance or the injection distance. Quite naturally also list-decoding algorithms were developed for Grassmannian codes (constructed either by linearlized polynomials or as lifted rank-metric codes), e.g =-=[2, 59, 60, 73, 74, 75, 76, 85, 111]-=-. Also, this direction of research has many problems for future research. Research problem 56. Suggest new classes of large Grassmannian codes, subspace codes (with the subspace distance or the inject...

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... in Fqn. Cyclic codes have a nice automorphism group. But, there are other automorphisms which can be forced on the code. One example is the use of the Frobenius mapping which was used for example in =-=[18]-=-. Some automorphisms of constant dimension codes were studied in [108]. Constructions for small dimensions might be attractive in this context. Interesting codes admitting some automorphisms were cons...

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... making use of necklaces in the Hamming graph. 14.2 Complements Complements of binary codewords and binary codes are used as a tool in various aspects of coding theory. The q-analog was considered in =-=[19]-=-. Various related problems concerning complements of subspaces over Fq were considered before, e. g. [30, 31, 32]. Definition 1. Let U be a subset of Pq(n) and let Uk := U ∩Gq(n, k). We say that a fun...

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...into disjoint copies of t− (n, k, λ)q designs. Parallelism in projective geometry is a large set and this topic will be discussed separately in Section 13. Braun, Kohnert, Österg̊ard, and Wassermann =-=[21]-=- presented a large set of 2 − (8, 3, 21)2 designs. This large set consists of three disjoint 2− (8, 3, 21)2 designs. Research problem 65. Find more large sets of t− (n, k, λ)q designs. 11 q-Covering D...

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...ing results. Lexicodes in the Grassmannian and their search were discussed in [96]. Codes which are able to correct also errors in coordinates (such as deletion or localized errors) are considered in =-=[24, 101]-=-. A question on the size of equidistant Grassmannian codes was asked in [50]. In an equidistant code, any two codewords have the same distance, which is clearly the minimum distance of the code. This ...

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...in [49] how to use the properties of subspace transversal design to obtain better bounds on Aq(n, δ, k) with codes which contains CMRD. These properties were also used to construct q-covering designs =-=[44]-=- and parallelisms [45] and they probably can be used for constructions of other related structures. Research problem 61. Find new t− (n, k, λ)q designs with new parameters. Research problem 62. Find q...

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... properties of subspace transversal design to obtain better bounds on Aq(n, δ, k) with codes which contains CMRD. These properties were also used to construct q-covering designs [44] and parallelisms =-=[45]-=- and they probably can be used for constructions of other related structures. Research problem 61. Find new t− (n, k, λ)q designs with new parameters. Research problem 62. Find q-analogs for other kno...

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...all the (k + 1)-dimensional subspaces? Research problem 97. What is the length of the longest path in G(PSq (2k + 1)) which contains only k-dimensional subspaces and (k + 1)-dimensional subspaces? In =-=[46]-=- it is shown that for any given q and k = 1 or k = 2 there exists a Hamiltonian cycle in the middle levels of Pq(2k + 1). The method is using cyclic shifts of subspaces in a modification of a similar ...

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...ly of codes might deserve further attention in the future. Another line of research for Grassmannain codes is based on Schubert calculus and Plücker coordinates. These were considered for example in =-=[57, 111, 113]-=- and their further research might lead to new interesting results. Lexicodes in the Grassmannian and their search were discussed in [96]. Codes which are able to correct also errors in coordinates (su...

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...s of a very special interest. This value has a special interest since (n, k, k)q codes have applications as byte-correcting codes [43, 65]. Decoding of such constant dimension codes was considered in =-=[58, 77]-=- The known upper and lower bounds on Aq(n, k, k) are summarized in the following theorems. The first three well-known theorems can be found in [50]. Theorem 8. If k divides n then Aq(n, k, k) = qn−1 q...

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...[50]. In an equidistant code, any two codewords have the same distance, which is clearly the minimum distance of the code. This problem is highly connected to problems in extremal combinatorics, e.g. =-=[53, 61]-=-. Some work in this direction was done lately in [62, 63, 64]. Finally, there are other related coding problems in the Grassmannian. For example, the intersection size of balls around codewords has so...

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...same distance, which is clearly the minimum distance of the code. This problem is highly connected to problems in extremal combinatorics, e.g. [53, 61]. Some work in this direction was done lately in =-=[62, 63, 64]-=-. Finally, there are other related coding problems in the Grassmannian. For example, the intersection size of balls around codewords has some interesting applications [118], where the case for the int...

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...same distance, which is clearly the minimum distance of the code. This problem is highly connected to problems in extremal combinatorics, e.g. [53, 61]. Some work in this direction was done lately in =-=[62, 63, 64]-=-. Finally, there are other related coding problems in the Grassmannian. For example, the intersection size of balls around codewords has some interesting applications [118], where the case for the int...

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...same distance, which is clearly the minimum distance of the code. This problem is highly connected to problems in extremal combinatorics, e.g. [53, 61]. Some work in this direction was done lately in =-=[62, 63, 64]-=-. Finally, there are other related coding problems in the Grassmannian. For example, the intersection size of balls around codewords has some interesting applications [118], where the case for the int...

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...ance or the injection distance. Quite naturally also list-decoding algorithms were developed for Grassmannian codes (constructed either by linearlized polynomials or as lifted rank-metric codes), e.g =-=[2, 59, 60, 73, 74, 75, 76, 85, 111]-=-. Also, this direction of research has many problems for future research. Research problem 56. Suggest new classes of large Grassmannian codes, subspace codes (with the subspace distance or the inject...

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...ance or the injection distance. Quite naturally also list-decoding algorithms were developed for Grassmannian codes (constructed either by linearlized polynomials or as lifted rank-metric codes), e.g =-=[2, 59, 60, 73, 74, 75, 76, 85, 111]-=-. Also, this direction of research has many problems for future research. Research problem 56. Suggest new classes of large Grassmannian codes, subspace codes (with the subspace distance or the inject...

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...If the code is constructed by the Multilevel Construction then the encoding is slightly more complicated and it was described in [69]. Finally, encoding of all the Grassmannian space was described in =-=[80, 95]-=-. Research problem 60. Find better encoding algorithms for subspace codes, the Grassmannian Gq(n, k), or the projective space Pq(n). 10 Designs over GF(q) q-analog of Steiner systems are one type of q...

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...s a q-analog of the well-known Turán design [34, 35]. The dual subspaces of the subspaces in a blocking set form a q-covering design Cq(n+1, n− t, n− s). Blocking sets were considered for example in =-=[82, 83]-=-. We note that blocking sets have also some different definitions (and maybe more popular definitions which define other structures which are not q-coverings). Similarly, to the case of error-correcti...