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Cyclic codes from some monomials and trinomials. (English) Zbl 1306.94114
Let \(s^\infty=(s_i)_{i=0}^\infty\) be a sequence in \(\text{GF}(q)\) of period \(q^m-1\) of the form \[ s_i=\sum_{j=0}^{q^m-2}c_j\alpha^{ij},\quad i\geq 0, \] where \(c_j\in\text{GF}(q^m)\) and \(\alpha\) is a primitive element of \(\text{GF}(q^m)\). The minimal polynomial of \(s^\infty\) is given by \[ \mathbb M_s(x)=\prod_{\substack{ 0\leq i\leq q^m-2\\ c_i\neq 0}}(1-\alpha^ix). \] In this paper, sequences of the form \[ s_i=\text{Tr}_{q^m/q}(f(\alpha^i+1)) \] are considered, where \(f\) is one of the following monomials and trinomial.
\[ q=2, \quad f(x)=x^{2^m-2}; \] \[ q \text{ odd},\quad f(x)=x^{q^\kappa+1}, \text{where } m/\text{gcd}(m,\kappa) \text{ is odd}; \] \[ q=3,\quad f(x)=x^{10}-ux^6-u^2x^2\in\text{GF}(3^m)[x]; \] \[ q \text{ odd},\quad f(x)=x^{(q^h-1)/(q-1)}, \text{where } 1\leq h\leq \lfloor m/2\rfloor; \] \[ q=3,\quad f(x)=x^{(3^h+1)/2}, \text{where } 3\leq h\leq \lfloor m/2\rfloor. \] In each of these cases, the minimal polynomial \(\mathbb M_s(x)\) of the sequence \(s^\infty\) is determined. The polynomial \(\mathbb M_s(x)\) generates a \(q\)-ary cyclic code \(\mathcal C_s\) of length \(q^m-1\). In each of the above cases, the dimension of \(\mathcal C_s\) is determined and bounds for the minimum distance \(d\) of \(\mathcal C_s\) are given. In most cases, the upper bound for \(d\) is the sphere packing bound and the lower bound for \(d\) is the BCH bound. It is observed through numerous examples that the code \(\mathcal C_s\) can be optimal or near optimal. It is pointed out that for a monomial \(f\), the minimal polynomial of the sequence \(s_i=\text{Tr}_{q^m/q}(f(\alpha^i+1))\) is related to that of the sequence \(\check{s}_i=\text{Tr}_{q^m/q}(f(\alpha^i+1)-f(\alpha^i))\) in a simple way. The paper contains several open questions inviting readers to improve the lower bound for the minimum distance for some of the codes considered in the paper and determine the generator polynomial and the parameters of the code \(\mathcal C_s\) for other suggested choices of \(f\).
Reviewer’s remark: In Eq. (5.3), \(v\alpha^t\) should be \((u^2+u^{3^{m-1}}-1)\alpha^t\). In Eq. (5.8), \(\mathbb N(J,t)\) equals \(\binom{J-1}{t-1}\).

MSC:
94B15 Cyclic codes
94B05 Linear codes, general
11T06 Polynomials over finite fields
11T71 Algebraic coding theory; cryptography (number-theoretic aspects)
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