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... establish bounds, or studies on the complexity reduction by coordinate permutation. Among them we can cite [11]–[26]. Recent results on the complexity of convolutional codes are reported in [17] an=-=d [27]-=-. For a complete list of references on the subject, see [28]. B. Minimal Span Generator Matrix (MSGM) Matrices A procedure for practically computing the state and branch profile of a code has been pre...

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...er distance spectrum than the best known PCCs with same code rate and trellis complexity is presented. I. INTRODUCTION ACONVOLUTIONAL code can be represented by a semi-infinite trellis consisting, after a short transient, of concatenated copies of a topological structure called trellis module. A trellis module M for a rate R = k/n (i.e., a (n, k)) convolutional code C consists of n′ ≤ n trellis sections (from depth 0 to depth n′), 2νt states at depth t, 2bt edges emanating from each state at depth t, and lt bits labeling each edge from depth t to depth t+1 (for 0 ≤ t ≤ n′−1). McEliece and Lin [1] stated that the computational effort required by the Viterbi algorithm to decode a convolutional code is proportional to the total number of edge symbols per information bit in the trellis module representing the code. This is said to be the trellis complexity of the module M , denoted by TC(M). There can be many trellis modules describing the same code. The trellis complexity of the conventional trellis module for the (n, k) convolutional code C with memory order ν, denoted by Mconv, is given by TC(Mconv) = (n/k) 2ν+k symbols per bit. Punctured convolutional codes (PCCs) [1] form a special c...

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...et G(D) be a polynomial generator matrix for an (n0, k0, ν) convolutional code C with memory ν. Denote by H(D) a corresponding parity-check matrix. Both G(D) and H(D) are assumed to be canonical [4], =-=[5]-=-. In this case, the code-trellis module associated with G(D) and the error-trellis modules [6] associated with the syndrome former HT (D) (T means transpose) have 2ν states, where the obvious realizat...

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...nary syndrome trellis representation. In general, this comes at the cost of increasing the state complexity from s = ν to s = ν + 1, that is, the number of states at each level of the trellis doubles =-=[11]-=-. In Section II, we show that a binary syndrome trellis can be realized with state complexity s = ν, that is, at most 2ν different states, if the parity-check polynomials fulfill certain conditions. T...

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... the Viterbi algorithm to decode C consists of a concatenation of infinitely many copies of the trellis modules M. The trellis complexity of the module M for the code C, denotedbyTC(M), is defined as =-=[16]-=- TC(M) = 1 n k ′ ∑−1 lt2νt+bt (1) symbols per bit. In particular, TC(Mconv) = (n/k)2ν+k symbols per bit. The “minimal” trellis module, ˜M, for convolutional codes was developed in [16, 17]. This “mini...

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...nsider trellis representations for the constituent codes other than the conventional trellis usually adopted. For nonrecursive convolutional encodes, there is a trellis structure, the minimal trellis =-=[6]-=-, which represents the coded sequences minimally under various complexity measures. The interest on the minimal trellis representation comes from its good error performance versus decoding complexity ...

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... Viterbi decoding of rate-k/n convolutional codes over GF(q) (abbreviated as SZ method or SZ construction). They constructed the so-called BCJR trellis [2] and have given the “minimal” trellis module =-=[14]-=- for a given convolutional code. Here, the term “minimal” means that the number of vertices at each depth is minimum. The minimal trellis module also minimizes the number of trellis symbols per encode...

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...eps: In step 1, all the edges are efficiently calculated by a special coset trellis. In step 2, the decoding results of step 1 are applied to a simplified convolutional code trellis. McEliece and Lin =-=[17]-=- recently described a decoding algorithm. As a particular example they showed that decoding of the code gI in Example 1 can be carried out with a total of 104 additions per information bit. Ascetic co...

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...in the number of transmit antenas. However, for the two cases considered, the degradation is still very small in terms of SNR. IV. ANALYSIS OF THE COMPUTATIONAL COMPLEXITY McEliece and Lin defined in =-=[19]-=- the trellis complexity for a convolutional code, which is directly related to the computational effort required by an algorithm like Viterbi [20] or BCJR [21] to decode one bit, and is a function of ...