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...uctions presented in this paper are performed algebraically and not by computational search. 1 Introduction Much effort have been paid in order to construct good quantum error-correcting codes (QECC) =-=[4, 9, 22, 24, 25, 36, 44]-=- as well as quantum convolutional codes with good parameters [1–3, 13–16,27, 37, 38]. On the other hand, the investigation of the class of (classical) convolutional codes and their corresponding prope...

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...uctions presented in this paper are performed algebraically and not by computational search. 1 Introduction Much effort have been paid in order to construct good quantum error-correcting codes (QECC) =-=[4, 9, 22, 24, 25, 36, 44]-=- as well as quantum convolutional codes with good parameters [1–3, 13–16,27, 37, 38]. On the other hand, the investigation of the class of (classical) convolutional codes and their corresponding prope...

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... β is a primitive 2nth root of unity, then the minimum distance dC of C satisfies dC ≥ d. 3 Classical Convolutional Codes The class of (classical) convolutional codes is a well-studied class of codes =-=[2, 3, 12, 18, 19, 39]-=-. We assume the reader is familiar with the theory of convolutional codes. For more details, see [19]. Recall that a polynomial encoder matrix G(D) = (gij) ∈ Fq[D] k×n is called basic if G(D) has a po...

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...In this paper, we utilize the class of negacyclic codes [5–8, 10, 23] in order to construct classical and quantum MDS convolutional codes. More precisely, we apply the famous method proposed by Piret =-=[39]-=- (generalized recently by Aly et al. [2]), which consists in the construction of (classical) convolutional codes derived from block codes. An advantage of our techniques of construction lie in the fac...

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...uctions presented in this paper are performed algebraically and not by computational search. 1 Introduction Much effort have been paid in order to construct good quantum error-correcting codes (QECC) =-=[4, 9, 22, 24, 25, 36, 44]-=- as well as quantum convolutional codes with good parameters [1–3, 13–16,27, 37, 38]. On the other hand, the investigation of the class of (classical) convolutional codes and their corresponding prope...

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...assical) convolutional codes and their corresponding properties as well as constructions of maximum-distance-separable (MDS) convolutional codes (i.e., codes attaining the generalized Singleton bound =-=[41]-=-) have also been presented in the literature [11, 12, 17, 26–34,39, 41–43]. In this paper, we utilize the class of negacyclic codes [5–8, 10, 23] in order to construct classical and quantum MDS convol...

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... β is a primitive 2nth root of unity, then the minimum distance dC of C satisfies dC ≥ d. 3 Classical Convolutional Codes The class of (classical) convolutional codes is a well-studied class of codes =-=[2, 3, 12, 18, 19, 39]-=-. We assume the reader is familiar with the theory of convolutional codes. For more details, see [19]. Recall that a polynomial encoder matrix G(D) = (gij) ∈ Fq[D] k×n is called basic if G(D) has a po...

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...uctions presented in this paper are performed algebraically and not by computational search. 1 Introduction Much effort have been paid in order to construct good quantum error-correcting codes (QECC) =-=[4, 9, 22, 24, 25, 36, 44]-=- as well as quantum convolutional codes with good parameters [1–3, 13–16,27, 37, 38]. On the other hand, the investigation of the class of (classical) convolutional codes and their corresponding prope...

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...s 5.5 and 5.6, the result follows. 6 New Quantum MDS-Convolutional codes As in the classical case, the construction of MDS quantum convolutional codes is a difficult task. This task is performed in =-=[3, 13, 14, 16]-=- but only in [3, 14] the constructions are made algebraically. Here, we propose the construction of MDS convolutional stabilizer codes derived from the convolutional codes constructed in Section 5. To...

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...more details, see [19]. Recall that a polynomial encoder matrix G(D) = (gij) ∈ Fq[D] k×n is called basic if G(D) has a polynomial right inverse. A basic generator matrix is called reduced (or minimal =-=[18, 32, 43]-=-) if the overall constraint length γ = k∑ i=1 γi, where γi = max1≤j≤n{deg gij}, has the smallest value among all basic generator matrices (in this case the overall constraint length γ will be called t...

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...n). The minimal polynomial (over Fq2) of β j ∈ Fq2m is denoted byM (j)(x) and it is given byM (j)(x) = ∏ j∈Ci (x−βj). The dimension of C is given by n − |Z|. The BCH bound for Constacyclic codes (see =-=[5, 23]-=-) asserts that is C is a q2-ary negacyclic code of length n with 2 generator polynomial g(x) and if g(x) has the elements {β2i+1|0 ≤ i ≤ d − 2} as roots, where β is a primitive 2nth root of unity, the...

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...uctions presented in this paper are performed algebraically and not by computational search. 1 Introduction Much effort have been paid in order to construct good quantum error-correcting codes (QECC) =-=[4, 9, 22, 24, 25, 36, 44]-=- as well as quantum convolutional codes with good parameters [1–3, 13–16,27, 37, 38]. On the other hand, the investigation of the class of (classical) convolutional codes and their corresponding prope...

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...s 5.5 and 5.6, the result follows. 6 New Quantum MDS-Convolutional codes As in the classical case, the construction of MDS quantum convolutional codes is a difficult task. This task is performed in =-=[3, 13, 14, 16]-=- but only in [3, 14] the constructions are made algebraically. Here, we propose the construction of MDS convolutional stabilizer codes derived from the convolutional codes constructed in Section 5. To...

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...uctions presented in this paper are performed algebraically and not by computational search. 1 Introduction Much effort have been paid in order to construct good quantum error-correcting codes (QECC) =-=[4, 9, 22, 24, 25, 36, 44]-=- as well as quantum convolutional codes with good parameters [1–3, 13–16,27, 37, 38]. On the other hand, the investigation of the class of (classical) convolutional codes and their corresponding prope...

Citation Context

... β is a primitive 2nth root of unity, then the minimum distance dC of C satisfies dC ≥ d. 3 Classical Convolutional Codes The class of (classical) convolutional codes is a well-studied class of codes =-=[2, 3, 12, 18, 19, 39]-=-. We assume the reader is familiar with the theory of convolutional codes. For more details, see [19]. Recall that a polynomial encoder matrix G(D) = (gij) ∈ Fq[D] k×n is called basic if G(D) has a po...

Citation Context

...more details, see [19]. Recall that a polynomial encoder matrix G(D) = (gij) ∈ Fq[D] k×n is called basic if G(D) has a polynomial right inverse. A basic generator matrix is called reduced (or minimal =-=[18, 32, 43]-=-) if the overall constraint length γ = k∑ i=1 γi, where γi = max1≤j≤n{deg gij}, has the smallest value among all basic generator matrices (in this case the overall constraint length γ will be called t...

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...gacyclic codes [5–8, 10, 23] in order to construct classical and quantum MDS convolutional codes. More precisely, we apply the famous method proposed by Piret [39] (generalized recently by Aly et al. =-=[2]-=-), which consists in the construction of (classical) convolutional codes derived from block codes. An advantage of our techniques of construction lie in the fact that all new (classical and quantum) c...

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... Theorem 6.2 [3] (Quantum Singleton bound) The free distance of an [(n, k, µ; γ, df )]q Fq2-linear pure convolutional stabilizer code is bounded by df ≤ n− k 2 (⌊ 2γ n+ k ⌋ + 1 ) + γ + 1. 8 Lemma 6.3 =-=[20]-=- Let n = q2 + 1, where q ≡ 1 (mod 4) is a power of an odd prime and suppose that s = n/2. If C is a q2-ary negacyclic code of length n with defining set Z = ∪δi=0Cs−2i, where 0 ≤ δ ≤ (q − 1)/2, then C...

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...n). The minimal polynomial (over Fq2) of β j ∈ Fq2m is denoted byM (j)(x) and it is given byM (j)(x) = ∏ j∈Ci (x−βj). The dimension of C is given by n − |Z|. The BCH bound for Constacyclic codes (see =-=[5, 23]-=-) asserts that is C is a q2-ary negacyclic code of length n with 2 generator polynomial g(x) and if g(x) has the elements {β2i+1|0 ≤ i ≤ d − 2} as roots, where β is a primitive 2nth root of unity, the...

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...ribed. 2 Negacyclic codes The class of negacyclic codes [5–8, 10, 21, 23] have been studied in the literature. This class of codes are a particular class of a more general class of constacyclic codes =-=[8]-=-. In this section we review the basic concepts of these codes. Throughout this paper, we always assume that q is a power of an odd prime, Fq is a finite field with q elements and n is a positive integ...

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...s 5.5 and 5.6, the result follows. 6 New Quantum MDS-Convolutional codes As in the classical case, the construction of MDS quantum convolutional codes is a difficult task. This task is performed in =-=[3, 13, 14, 16]-=- but only in [3, 14] the constructions are made algebraically. Here, we propose the construction of MDS convolutional stabilizer codes derived from the convolutional codes constructed in Section 5. To...