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...or matrix has the LR property. It follows We have . The minimal trellis of is depicted in Fig. 1(b). In general, the problem of finding the best permutation for complexity minimization is NP-complete =-=[25], [2-=-6]. D. Sectionalization So far, we have considered an “atomic” representation of , i.e., a bit at every time . For clearness, in the following we will denote by , and , the complexity parameters o...

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...re of trellis complexity of a linear code [14], [5], [17]. However, there appears to be no standard coding-theoretic nomenclature for this notion. It has been called the state complexity of a code in =-=[10]-=-, but the use of this term there conflicts slightly with its use in [17]. So to avoid ambiguity, we will give it a new name here — trellis-width — which acknowledges its roots in trellis complexity. T...

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...e may therefore lead to useful results for each discipline. We point out that the possibility of connection between binary decision diagrams and trellises was noted in passing by Horn and Kschischang =-=[42], who wrot-=-e that "blockcode trellises appear to be closely related to graphs called binary decision diagrams that are used to represent Boolean functions." However, to the best of our knowledge, this ...

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...oblem above, i.e., the problem of finding a permutation that minimizes a given element of the state complexity profile for a general linear block code, was shown to be NP-hard by Horn and Kschischang =-=[18]-=-. Optimum (uniformly efficient) permutations are known for some codes, e.g., the [24,12,8] Golay code [15], the [48,24,12] quadratic residue code [2,10], and the [16,7,6] lexicode [23]. The most gener...

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...the problems that we study here. The first results of this type were by Berlekamp, McEliece, and van Tilborg [14], and many other examples are given by Vardy [15] and Sudan [16]. Horn and Kschischang =-=[17]-=- and Kschischang and Sorokine [18] have shown NP-completeness of certain problems of minimizing the trellis complexity of an error-correcting code. Regarding related work in complexity theory, one of ...

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...ces and Engineering Research Council (NSERC) of Canada. 2 It now matters which measure of trellis complexity is to be minimized, as the coordinate permutation of C that yields a minimal trellis with, say, the least state-complexity need not be the same as the coordinate permutation that yields a minimal trellis with the smallest number of states. The prior literature has most often focused on the problem of finding the coordinate permutation of a given code that minimizes the state-complexity of the resulting minimal trellis, and it has repeatedly been conjectured that this problem is NP-hard [5], [6], [11, Section 5]. To put it another way, the following decision problem was conjectured to be NP-complete: Problem: Trellis State-Complexity Let Fq be a fixed finite field. Instance: An m×n generator matrix for a linear code C over Fq, and an integer w > 0. Question: Is there a coordinate permutation of C that yields a code C′ whose minimal trellis has state-complexity at most w? This decision problem was called “Maximum Partition Rank Permutation” in [5], and “Maximum Width” in [6]. Forney [4] has referred to the resolution of the aforementioned conjecture as the “only significant open ...

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... such a case, we will write C ≡ C ′. The equivalence class of codes equivalent to C will be denoted by [C]. III. TRELLIS COMPLEXITY AND MINORS We will define trellis state-complexity via the rather useful notion of the connectivity function of a linear code C. This is the function λC : 2E(C) → Z defined by λC(J) = dim(C|J ) + dim(C|Jc) − dim(C), (5) for each J ⊂ E(C). It is obvious that for any J ⊂ E(C), we have λC(J) ≥ 0 and λC(J) = λC(Jc). Observe also that λC(∅) = λC(E(C)) = 0. Let C be a linear code over Fq , with label sequence αC = (α1, α2, . . . , αn). The state-complexity profile [5], [9] of C is the sequence s(C) = (s0(C), s1(C), . . . , sn(C)) defined as follows: s0(C) = sn(C) = 0, and for 1 ≤ i ≤ n − 1, si(C) = λC({α1, α2, . . . , αi}). (6) The quantities si(C) determine the size of the minimal trellis of C — the number of vertices (states) at time i in the minimal trellis is precisely qsi(C). The state-complexity of (the minimal trellis of) the code C is defined to be smax(C) = maxi∈[n] si(C). As was noted by Muder [15], equivalent codes C and C ′ may have very different minimal trellises. Therefore, we may have s(C) 6= s(C′), and even smax(C) 6= smax(C′). It thus makes se...