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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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