On some properties of the hyper-Kloosterman codes
Let \(m,r\geq 2\) be integers, \(p\) a prime and \(q=p^r\). In a previous paper, the author and \textit{T. Hiramatsu} [Appl. Algebra Eng. Commun. Comput. 12, 381--390 (2001; Zbl 0987.11077)] introduced the hyper-Kloosterman codes \(C_m(q)\). This code is contained in \({\mathbb F}_p^{(q-1)^{m-1}}\) and a codeword is obtained from the image of \(\mathbf a\in {\mathbb F}_q^m\) via the map \({\mathbf a}:\mapsto \{\text{Tr}(\mathbf a,\mathbf x)\}_{\{{\mathbf x}\in ({\mathbb F}_q^*)^{m-1}\}}\), where \[ \text{Tr}(\mathbf a,\mathbf x):=\text{ tr}(a_1x_1+\ldots+a_{m-1}x_{m-1}+ a_m(x_1\ldots x_{m-1})^{-1}) \] and \(\text{ tr}\) is the trace from \({\mathbb F}_q\) to \({\mathbb F}_p\). These codes generalize the Kloosterman codes, they are trace codes and define a new type of quasi-cyclic code; a key property proved here is that there exists a code \(\tilde C_m(q)\) such that \(\text{ tr}(\tilde C_m(q))=C_m(q)\) in such a way that the dimension of \(C_m(q)\) can be computed. Another result in this paper concerns the minimum distance \(d\) of \(C_m(q)\) whenever \((p-1)| m\). In the aforementioned paper, the weight of the codewords of \(C_m(q)\) have been computed: one concludes that the \(d\) equals the biggest one of \((q-1)^m\) and the hyper-Kloosterman sums \(K_m(\mathbf a,q)\) (\(\mathbf a\in ({\mathbb F}_q^*)^m\)). Set \(e(x):=e^{2\pi ix/p}\). With the above notation, \[ K_m(\mathbf a,q):=\sum_{\mathbf x\in ({\mathbb F}^{*})^{m-1}} e(\text{Tr}(\mathbf a,\mathbf x))\, ; \] in general it is difficult to compute this number but there is an upper bound for it, namely Deligne's bound: \(K_m(\mathbf a,q)\leq mq^{(m-1)/2}\). By means of these facts the author computes (or obtains lower bounds for) the minimum distance of hyper-Kloosterman codes for certain values of \(m\) and \(q\). The author shows some examples of (binary) codes, as those discussed here, which are efficient according to the Gilbert-Varshamov bound; finally, he writes an interesting algorithm to compute Kloosterman sums.
scientific article; zbMATH DE number 2064095