On Rank and Kernel of ℤ4-Linear Codes

New characterisations of the Nordstrom–Robinson codes

“…In their paper, Hammons et al also gave an explanation to one of the outstanding problems in coding theory, that the weight enumerators of the non-linear Kerdock codes and the Preparata codes satisfy the MacWilliams identities. The first member of both of these families is the well-known Nordstrom-Robinson code N , which is a non-linear (16,256,6) binary code with several interesting properties. It is optimal, in the sense that it is the largest possible binary code of length 16 with minimum distance 6, and it is twice as large as any linear binary code with the same length and minimum distance.…”

“…It is optimal, in the sense that it is the largest possible binary code of length 16 with minimum distance 6, and it is twice as large as any linear binary code with the same length and minimum distance. Moreover, Snover [40] proved that any binary (16,256,6) code is equivalent to the Nordstrom-Robinson code. Analogous properties also hold for the punctured Nordstrom-Robinson code, a non-linear (15,256,5) code.…”

“…Theorem 1.1. Any binary completely regular code of length m with minimum distance δ is equivalent to the Nordstrom-Robinson code, respectively, the punctured Nordstrom-Robinson code, if (m, δ) = (16,6) or (15,5). Moreover, such a code is completely transitive.…”

Kernels and ranks of cyclic and negacyclic quaternary codes

“…It is well known that the kernel of a binary code is the intersection of all maximal linear subspaces and that the code is the union of cosets of the kernel; see [9], [10] for details.…”

“…In [9], [10], various bounds are put on the rank and size of the kernel for arbitrary quaternary codes. In this work, these bounds are significantly refined for the cyclic case.…”