The canonical $\mathbb{Z}_{2^s}$-linear Hadamard codes are the Gray images of the $\mathbb{Z}_{2^s}$-additive codes. Their rank, the dimension of the binary linear span, was known in closed form only for $s\le 3$, and for $s \ge 4$ only through computer computations for small lengths. We give a method that computes the...
Linear complementary dual (LCD) codes are linear codes that intersect their dual trivially. We study double circulant (Euclidean) LCD codes over the finite chain ring $R=\mathbb F_2+u\mathbb F_2$, where $u^2=0$. Using the Chinese Remainder Theorem and the factorization of $x^n-1$, we derive an explicit enumeration form...
A subspace of $\mathbb{F}_q^n$ is called cyclically covering if the union of $\sigma^i(U)$ can cover the whole space $\mathbb{F}_q^n$, where $\sigma$ is the cyclic shift, $0 \leqslant i \leqslant n-1$. Let $h_q(n)$ be the largest possible co-dimension of a cyclically covering subspace of $\mathbb{F}_q^n$. We show that...
Xuan Wang, Minjia Shi· arXiv.org· 0 citations
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