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Linear codes from vectorial Boolean power functions. (English) Zbl 1465.94104

Let \(p=2\), \(q=p^m\), for every natural number \(l|m\) defining the trace function \(\text{Tr}_l^{m}(x)=\sum_{i=0}^{(\frac{m}l-1)l}{x^{p^i}}\) consider the linear code \[\mathcal{C}_D=\{c_u=(\text{Tr}_1^m(ud_1),\ldots,\text{Tr}_1^m(ud_n)\mid u\in \mathbb{F}_{2^m}\},\] for the defining set \(D=\{d_1,\ldots,d_n\}\subseteq \mathbb{F}_{2^m}\). The authors use the preimage of the trace \(\text{Tr}_l^{m}\) of some vectorial Boolean power functions as the defining sets where \(f(x)\) are power functions.
For \(m = 3l\), a class of three-weight linear codes is constructed by taking the inverse function \(f(x)=x^{-1}\). The weight distribution is completely determined by using known facts about trace functions.
Further, when \(l\) and 3 are coprime, by considering \(f(x) = x^{2^k}+1,\) it is shown that a class of four-weight linear codes and a class of two-weight codes can be constructed for \(m\) odd and \(\gcd(k, m)=1\) and for \(m\) even and \(k=\frac{m}{2}.\) At the end of the paper, it is shown that a class of simplex codes and a class of first-order Reed-Muller codes can also be obtained from this construction by choosing particular functions and defining sets.

MSC:

94B05 Linear codes (general theory)
11L03 Trigonometric and exponential sums (general theory)
11T71 Algebraic coding theory; cryptography (number-theoretic aspects)
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