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Reversible codes have a range of wide applications in cryptography, data storage, and communication systems. In this paper, we investigated reversible codes under the Rosenbloom-Tsfasman metric (RT-metric). First, some properties of reversible codes in the RT-metric were described. An essential condition for a reversible code to be a maximum distance separable code (MDS code, in short) in the RT-metric was established. A necessary condition for a binary self-dual code to be reversible was proven and the same was generalized for q-ary self-dual reversible codes. Several constructions for reversible RT-metric codes were provided in terms of their generator matrices. [ABSTRACT FROM AUTHOR]

In 2018, J. M. Gómez et al. showed that the problem of finding the packing number ρ(F2(Pn)) of the 2-token graph F2(Pn) of the path Pn of length n ≥ 2 is equivalent to determining the maximum size of a binary code S' of constant weight w = 2 that can correct a single adjacent transposition. By determining the exact value of ρ(F2(Pn)), they proved a conjecture of Rob Pratt. In this paper, we study a related problem, which consists of determining the packing number ρ(F3(Pn)) of the graph F3(Pn). This problem corresponds to the Sloane's problem of finding the maximum size of S' of constant weight w = 3 that can correct a single adjacent transposition. Since the maximum packing set problem is computationally equivalent to the maximum independent set problem, which is an NP-hard problem, then no polynomial time algorithms are expected to be found. Nevertheless, we compute the exact value of ρ(F3(Pn)) for n ≤ 12, and we also present some algorithms that produce a lower bound for ρ(F3(Pn)) with 13 ≤ n ≤ 44. Finally, we establish an upper bound for ρ(F3(Pn)) with n ≥ 13. [ABSTRACT FROM AUTHOR]

In this paper, we introduced double cyclic codes over Rr × Rs, where R = Z2 + uZ2 = {0, 1, u, 1 + u} is the ring with four elements and u² = 0. We first determined the generator polynomials of R-double cyclic codes for odd integers r and s, then gave the generators of duals of free double cyclic codes over Rr × Rs. By defining a linear Gray map, we looked at the binary images of R-double cyclic codes and gave several examples of optimal parameter binary linear codes obtained from R-double cyclic codes. Moreover, we studied self-dual R-double cyclic codes and presented an example of a self-dual R-double cyclic code. [ABSTRACT FROM AUTHOR]

Let da(n, 5) and dl(n, 5) be the minimum weights of optimal binary [n, 5] linear codes and linear complementary dual (LCD) codes, respectively. This article aims to investigate dl(n, 5) of some families of binary [n, 5] LCD codes when n = 31s + t ≥ 14 with s integer and t ∈ {2, 8, 10, 12, 14, 16, 18}. By determining the defining vectors of optimal linear codes and discussing their reduced codes, we classify optimal linear codes and calculate their hull dimensions. Thus, the non-existence of these classes of binary [n, 5, da(n, 5)] LCD codes is verified, and we further derive that dl(n, 5) = da(n, 5) − 1 for t, 16 and dl(n, 5) = 16s + 6 = da(n, 5) − 2 for t = 16. Combining them with known results on optimal LCD codes, dl(n, 5) of all [n, 5] LCD codes are completely determined. [ABSTRACT FROM AUTHOR]