This letter is concerned with incorrigible sets of binary linear codes. For a given binary linear code C, we represent the numbers of incorrigible sets of size up to ⌈3/2d - 1⌉ using the weight enumerator of C, where d is the minimum distance of C. In addition, we determine the incorrigible set enumerators of binary Golay codes G23 and G24 through combinatorial methods.

Linear codes with a few-weight have important applications in combinatorial design, strongly regular graphs and cryptography. In this paper, we first construct a class of Boolean functions with at most five-valued Walsh spectra, and determine their spectrum distribution. Then, we derive two classes of linear codes with at most six-weight from the new functions. Meanwhile, the length, dimension and weight distributions of the codes are obtained. Results show that both of the new codes are minimal and among them, one is wide minimal code and the other is a narrow minimal code and thus can be used to design secret sharing scheme with good access structures. Finally, some Magma programs are used to verify the correctness of our results.

In this letter, we consider the incorrigible sets of binary linear codes. First, we show that the incorrigible set enumerator of a binary linear code is tantamount to the Tutte polynomial of the vector matroid induced by the parity-check matrix of the code. A direct consequence is that determining the incorrigible set enumerator of binary linear codes is #P-hard. Then for a cycle code, we express its incorrigible set enumerator via the Tutte polynomial of the graph describing the code. Furthermore, we provide the explicit formula of incorrigible set enumerators of cycle codes constructed from complete graphs.

In this letter, we investigate the separating redundancy of binary linear codes. Using analytical techniques, we provide a general lower bound on the first separating redundancy of binary linear codes and show the bound is tight for a particular family of binary linear codes, i.e., cycle codes. In other words, the first separating redundancy of cycle codes can be determined. We also derive a deterministic and constructive upper bound on the second separating redundancy of cycle codes, which is shown to be better than the general deterministic and constructive upper bounds for the codes.

The central limit theorem (CLT) claims that the standardized sum of a random sequence converges in distribution to a normal random variable as the length tends to infinity. We prove the existence of a family of counterexamples to the CLT for d-tuplewise independent sequences of length n for all d=2,...,n-1. The proof is based on [n, k, d+1] binary linear codes. Our result implies that d-tuplewise independence is too weak to justify the CLT, even if the size d grows linearly in length n.

Generalized Minimum Distance (GMD) decoding is a well-known soft-decision decoding for linear codes. Previous research on GMD decoding focused mainly on unquantized AWGN channels with BPSK signaling for binary linear codes. In this paper, a study on the design of a 4-level uniform quantizer for GMD decoding is given. In addition, an extended version of a GMD decoding algorithm for a 4-level quantizer is proposed, and the effectiveness of the proposed decoding is shown by simulation.

In this paper, a study of a sufficient condition on the optimality of a decoded codeword of soft-decision decodings for binary linear codes is shown for a quantized case. A typical uniform 4-level quantizer for soft-decision decodings is employed for the analysis. Simulation results on the (64,42,8) Reed-Muller code indicates that the condition is effective for SN ratios at 3[dB] or higher for any iterative style optimum decodings.

In this paper, a study on the design and implementation of uniform 4-level quantizers for soft-decision decodings for binary linear codes is shown. Simulation results on quantized Viterbi decoding with a 4-level quantizer for the (64,42,8) Reed-Muller code show that the optimum stepsize, which is derived from the cutoff rate, gives an almost optimum error performance. In addition, the simulation results show that the case where the number of optimum codewords is larger than the one for a received sequence causes non-negligible degradation on error performance at high SN ratios of Eb/N0.

In this short correspondence, (1+uv)-constacyclic codes over the finite non-chain ring R[v]/(v2+v) are investigated, where R=F2+uF2 with u2=0. Some structural properties of this class of constacyclic codes are studied. Further, some optimal binary linear codes are obtained from these constacyclic codes.

An algorithm for augmenting a binary linear code is presented. The input to the code augmenting algorithm is (n,k,d) code C and the output is an (n,k*,d) augmented code C (k* k) satisfying C C and the Gilbert bound. The algorithm can be considered as an efficient implementation of the proof of Gilbert bound; for a given binary linear code C, the algorithm first finds a coset leader with the largest weight. If the weight of the coset leader is greater than or equal to the minimum distance of C, the coset leader is included to the basis of C.

In an iterative decoding algorithm, such as Chase Type-II decoding algorithm and its improvements, candidate codewords for a received vector are generated for test based on a bounded-distance decoder and a set of test error patterns. It is desirable to remove useless test error patterns in these decoding algorithms. This paper presents a sufficient condition for ruling out some useless test error patterns. If this condition holds for a test error patterns e, then e can not produce a candidate codeword with a correlation metric larger than those of the candidate codewords generated already and hence e is useless. This significantly reduces the decoding operations in Chase type-II decoding algorithm or decoding iterations in its improvements.

Subtrellises for low-weight codewords of binary linear block codes have been recently used in a number of trellis-based decoding algorithms to achieve near-optimum or suboptimum error performance with a significant reduction in decoding complexity. An algorithm for purging a full code trellis to obtain a low-weight subtrellis has been proposed by H.T. Moorthy et al. This algorithm is effective for codes of short to medium lengths, however for long codes, it becomes very time consuming. This paper investigates the structure and complexity of low-weight subtrellises for binary linear block codes. A construction method for these subtrellises is presented. The state and branch complexities of low-weight subtrellises for Reed-Muller codes and some extended BCH codes are given. In addition, a recursive algorithm for searching the most likely codeword in low-weight subtrellis-based decoding algorithm is proposed. This recursive algorithm is more efficient than the conventional Viterbi algorithm.

Motivated by the design of fault-tolerant multiprocessor interconnection networks, this paper considers the following problem: Given a positive integer t and a graph H, construct a graph G from H by adding a minimum number Δ(t, H) of edges such that even after deleting any t edges from G the remaining graph contains H as a subgraph. We estimate Δ(t, H) for the hypercube and torus, which are well-known as important interconnection networks for multiprocessor systems. If we denote the hypercube and the square torus on N vertices by QN and DN respectively, we show, among others, that Δ(t, QN) = O(tN log(log N/t + log 2e)) for any t and N (t 2), and Δ(1, DN) = N/2 for N even.