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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 2 𝑑 − 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 G 23 and G 24 through combinatorial methods.

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Given an odd prime 𝑞 and an integer 𝑚 ≤ 𝑞, a binary 𝑚𝑞 × 𝑞 2 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.