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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 $x=\left({x}_{1},{x}_{2},\cdots ,{x}_{n}\right)\in V$ and $r\in R$, the complete weight of $x$ is defined by ${n}_{r}\left(x\right)=|\left\{i\mid {x}_{i}=r\right\}|$. Let ${a}_{0}=0$ and ${a}_{r}$ be a positive real number for any nonzero $r\in R$. A general weight function $w\left(x\right)$ is defined by

$w\left(x\right)=\sum _{r\in R}{a}_{r}{n}_{r}\left(x\right)·$

For any linear $\left[n,k,d\right]$ 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 $w\left(x\right)$ be a general weight function on $C$. Then

$\left[\left(d-1\right)/A\right]\le n-{log}_{|R|}|C|,$

where $A=max\left\{{a}_{r}\mid r\in R\right\}$ and $\left[b\right]$ is the integer part of $b$.

Finally, the author presents some applications of his result to codes over ${ℤ}_{l}$ (and in particular ${ℤ}_{4}\right)$ for the special weight functions corresponding to the Hamming, Lee and Euclidean weights.

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