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Singleton bounds for codes over finite rings. (English) Zbl 0979.94052
Let $R$ be a finite commutative quasi-Froebenius (QF) ring and let $V=R^n$ be the free module of rank $n$ consisting of all $n$-tuples of elements of $R$. A code $C$ of length $n$ over $R$ is an $R$-submodule of $V$. For every $x=(x_1,x_2, \dots,x_n) \in V$ and $r\in R$, the complete weight of $x$ is defined by $n_r(x)= |\{i \mid x_i=r\} |$. Let $a_0=0$ and $a_r$ be a positive real number for any nonzero $r\in R$. A general weight function $w(x)$ is defined by $$w(x)= \sum_{r\in R}a_r n_r(x).$$ For any linear $[n,k,d]$ code over a finite field, the Singleton bound says that $d\le n-k+1$. The main result here is the following generalization of the Singleton bound for a general weight function over $R$. Theorem. Let $C$ be a code with minimum distance $d$ over a finite commutative QF ring $R$. Let $w(x)$ be a general weight function on $C$. Then $$\bigl[(d-1)/ A\bigr]\le n-\log_{|R|} |C|,$$ where $A=\max\{a_r \mid r\in R\}$ and $[b]$ is the integer part of $b$. Finally, the author presents some applications of his result to codes over $\bbfZ_l$ (and in particular $\bbfZ_4)$ for the special weight functions corresponding to the Hamming, Lee and Euclidean weights.

94B65Bounds on codes
16L60Quasi-Frobenius rings
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