cryptographic system developers

Abstract: We complete the determination of the minimum possible length of eight-dimensional binary self-orthogonal codes with given minimum distance. For dimension nine we give exact values and bounds for minimum distance at most $64$. By exploiting the geometric reformulation of linear codes as multisets of points, we fully redo the classification for dimension at most seven in a simplified and computer-free manner. As a by-product we prove the non-existence of $[93,9,44]_2$ codes and show several divisibility results for putative optimal nine-dimensional binary codes.

Abstract: The $\mathbb{Z}_{p^s}$-additive codes are subgroups of $\mathbb{Z}_{p^s}^n$. A $\mathbb{Z}_{p^s}$-linear generalized Hadamard (GH) code is a GH code over $\mathbb{Z}_p$ which is the Gray map image of a $\mathbb{Z}_{p^s}$-additive code. A recursive construction gives such codes $H_p^{t_1,\dots,t_s}$, of type $(n;t_1,\dots,t_s)$ and length $p^t$, and a partial classification of them is known. In this paper, we complete the classification of these codes for every prime $p$. The main tool is the derived code, obtained by puncturing the code outside a block of coordinates determined by its kernel. For a nonlinear code $H_p^{t_1,\dots,t_s}$ with $t_1\geq2$, the derived code is a $\mathbb{Z}_{p^{s-1}}$-linear GH code of the same family, although the mechanism for odd $p$ differs from the binary one. As a consequence, for $t\geq3$, there are $\mathcal P(t+1)-\mathcal P(t)$ nonequivalent such codes of length $p^t$ if $p$ is odd, and $\mathcal P(t+1)-\mathcal P(t)-1$ if $p=2$, where $\mathcal P(n)$ is the number of partitions of $n$. In particular, this completes the classification of the $\mathbb{Z}_{2^s}$-linear Hadamard codes. We also prove a conjecture on the codes with $t_1\geq2$ and $t_s=0$.

Abstract: The $\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-additive codes are subgroups of $\mathbb{Z}_2^{α_1}\times\mathbb{Z}_4^{α_2}\times\mathbb{Z}_8^{α_3}$, and a $\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-linear Hadamard code is the Gray map image of such a code. A recursive construction of $\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-additive Hadamard codes $\mathcal H^{t_1,t_2,t_3}$, with all $α_i\neq0$, $t_1\geq1$, $t_2\geq0$, and $t_3\geq1$, is known, as are the linearity, kernel dimension, and rank of the corresponding codes $H^{t_1,t_2,t_3}$ of length $2^t$, where $t+1=3t_1+2t_2+t_3$. Yet these invariants do not completely classify the family. Two infinite families of pairs of distinct types share the length, rank and kernel dimension, and for classified lengths $2^t$, $3\leq t\leq11$, such pairs were separated only by computer equivalence tests. In this paper, we introduce an equivalence invariant that resolves these cases. The kernel partitions the binary coordinates into blocks, two coordinates lying in the same block when every kernel word takes the same value in both; the \emph{kernel-block rank profile} is the multiset of the dimensions of the linear span punctured on these blocks. Unlike rank and kernel dimension, which are global, this invariant records how much of the span survives on each block. We compute it for the whole family: there are $2^{t_1+t_2+t_3-1}$ blocks, all of size $2^{2t_1+t_2}$, and the profile takes at most two values $t_2+\binom{t_1+2}{2}$ and $t_2+2+\binom{t_1+1}{2}$, whose difference is $t_1-1$; it is constant precisely when $t_1=1$. Hence, the profile recovers $t_1$, and the length and kernel dimension recover $t_2$ and $t_3$. Two codes of the family with the same length are therefore equivalent if and only if their types coincide, and the number of pairwise nonequivalent such codes of length $2^t$ is $\lfloor(t^2+6)/12\rfloor$ for every $t\geq3$.