A generalization of multi-twisted codes over finite fields, their Galois duals and Type II codes

Abstract

Let $\mathbb {F}_q$ denote the finite field of order q,  and let $\Lambda =(\lambda _1,\lambda _2,\cdots ,\lambda _\ell ),$ where $\lambda _1,\lambda _2,\cdots ,\lambda _\ell$ are non-zero elements of $\mathbb {F}_q.$ Let $n =m_1+m_2+\cdots +m_\ell ,$ where $m_1,m_2,\cdots ,m_\ell$ are arbitrary positive integers (not necessarily coprime to q). In this paper, we study algebraic structures of $\Lambda$-multi-twisted ($\Lambda$-MT) codes of length n and block lengths $(m_1,m_2,\cdots ,m_{\ell })$ over $\mathbb {F}_q$ and their Galois duals (i.e., their orthogonal complements with respect to the Galois inner product on $\mathbb {F}_{q}^n$). We develop generator theory for $\Lambda$-MT codes of length n over $\mathbb {F}_{q}$ and show that each $\Lambda$-MT code of length n over $\mathbb {F}_{q}$ has a unique nice normalized generating set. With the help of a normalized generating set, we explicitly determine the dimension and a generating set of the Galois dual of each $\Lambda$-MT code of length n over $\mathbb {F}_{q}.$ We also provide a trace description of all $\Lambda$-MT codes of length n over $\mathbb {F}_{q}$ by using the generalized discrete Fourier transform (GDFT), which gives rise to a method to construct these codes. We further provide necessary and sufficient conditions under which a Euclidean self-dual $\Lambda$-MT code of length n over $\mathbb {F}_{2^e}$ is a Type II code when $\lambda _i=1$ and $m_i = n_i2^a$ for $1\le i \le \ell ,$ where $a \ge 0$ is an integer and $n_1, n_2, \cdots , n_{\ell }$ are odd positive integers satisfying $n_1 \equiv n_2 \equiv \cdots \equiv n_{\ell }~ (\text {mod }4).$ Besides this, we obtain several linear codes with best-known and optimal parameters from 1-generator $\Lambda$-MT codes over $\mathbb {F}_{q},$ where $2 \le q \le 7.$ It is worth mentioning that these code parameters can not be attained by any of their subclasses (such as constacyclic and quasi-twisted codes) containing record breaker codes.