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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 $$d\leq n-k+1$$. 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 $\bigl[(d-1)/ A\bigr]\leq n-\log_{|R|} |C|,$ 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 $$\mathbb{Z}_l$$ (and in particular $$\mathbb{Z}_4)$$ for the special weight functions corresponding to the Hamming, Lee and Euclidean weights.

##### MSC:
 94B65 Bounds on codes 16L60 Quasi-Frobenius rings
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##### References:
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