Given an odd prime q and an integer m ≤ q, a binary mq × q2 quasi-cyclic parity-check matrix H(m, q) can be constructed for an array low-density parity-check (LDPC) code C (m, q). In this letter, we investigate the first separating redundancy of C (m, q). We prove that H (m, q) is 1-separating for any pair of (m, q), from which we conclude that the first separating redundancy of C (m, q) is upper bounded by mq. Then we show that our upper bound on the first separating redundancy of C (m, q) is tighter than the general deterministic and constructive upper bounds in the literature. For m=2, we further prove that the first separating redundancy of C (2, q) is 2q for any odd prime q. For m ≥ 3, we conjecture that the first separating redundancy of C (m, q) is mq for any fixed m and sufficiently large q.

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.

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.

The separating redundancy is an important property in the analysis of the error-and-erasure decoding of a linear block code. In this work, we investigate the separating redundancy of the duals of first-order generalized Reed-Muller (GRM) codes, a class of nonbinary linear block codes that have nice algebraic properties. The dual of a first-order GRM code can be specified by two positive integers m and q and denoted by R(m,q), where q is the power of a prime number and q≠2. We determine the first separating redundancy value of R(m,q) for any m and q. We also determine the second separating redundancy values of R(m,q) for any q and m=1 and 2. For m≥3, we set up a binary integer linear programming problem, the optimum of which gives a lower bound on the second separating redundancy of R(m,q).

The separating redundancy is an important concept in the analysis of the error-and-erasure decoding of a linear block code using a parity-check matrix of the code. In this letter, we derive new constructive upper bounds on the second separating redundancies of low-density parity-check (LDPC) codes constructed from projective and Euclidean planes over the field Fq with q even.

For an odd prime q and an integer m≤q, we can construct a regular quasi-cyclic parity-check matrix HI(m,q) that specifies a linear block code CI(m,q), called an improper array code. In this letter, we prove the minimum distance of CI(4,q) is equal to 10 for any q≥11. In addition, we prove the minimum distance of CI(5,q) is upper bounded by 12 for any q≥11 and conjecture the upper bound is tight.

Let G11 (resp., G12) be the ternary Golay code of length 11 (resp., 12). In this letter, we investigate the separating redundancies of G11 and G12. In particular, we determine the values of sl(G11) for l = 1, 3, 4 and sl(G12) for l = 1, 4, 5, where sl(G11) (resp., sl(G12)) is the l-th separating redundancy of G11 (resp., G12). We also provide lower and upper bounds on s2(G11), s2(G12), and s3(G12).

In this letter, the structural analysis of nonbinary cyclic and quasi-cyclic (QC) low-density parity-check (LDPC) codes with α-multiplied parity-check matrices (PCMs) is concerned. Using analytical methods, several structural parameters of nonbinary cyclic and QC LDPC codes with α-multiplied PCMs are determined. In particular, some classes of nonbinary LDPC codes constructed from finite fields and finite geometries are shown to have good minimum and stopping distances properties, which may explain to some extent their wonderful decoding performances.

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.

Based on the codewords of the [q,2,q-1] extended Reed-Solomon (RS) code over the finite field Fq, we can construct a regular binary γq×q2 matrix H(γ,q), where q is a power of 2 and γ≤q. The matrix H(γ,q) defines a regular low-density parity-check (LDPC) code C(γ,q), called a full-length RS-LDPC code. Using some analytical methods, we completely determine the values of s(H(4,q)), s(H(5,q)), and d(C(5,q)) in this letter, where s(H(γ,q)) and d(C(γ,q)) are the stopping distance of H(γ,q) and the minimum distance of C(γ,q), respectively.