A characterization of all semiprimitive irreducible cyclic codes in terms of their lengths (Q2280310)

Let \(q\) be a power of the prime number \(p\), and let \(n\) be a positive integer with \(\gcd(n,q)=1\). Let \(\mathbb{F}_{q}\) denote the finite field of order \(q\), and let \(\mathbb{F}_{q}^n\) denote the \(n\)-dimensional vector space consisting of all \(n\)-tuples over \(\mathbb{F}_{q}\). A cyclic code \(\mathcal{C}\) of length \(n\) over \(\mathbb{F}_{q}\) is defined as an \(\mathbb{F}_{q}\)-linear subspace of \(\mathbb{F}_{q}^n\), which is invariant under the cyclic shift of its codewords. Each cyclic code of length \(n\) over \(\mathbb{F}_{q}\) can also be viewed as an ideal of the quotient ring \(\mathbb{F}_{q}[x]/\langle x^n-1\rangle\) under the standard \(\mathbb{F}_{q}\)-linear vector space isomorphism from \(\mathbb{F}_{q}^n\) onto \(\mathbb{F}_{q}[x]/\langle x^n-1\rangle\), defined as \((a_0,a_1,\dots,a_{n-1}) \mapsto a_0+a_1x+a_2x^2+\dots+a_{n-1}x^{n-1}+\langle x^n-1\rangle \) for all \((a_0,a_1,a_2,\dots,a_{n-1}) \in \mathbb{F}_{q}^n\). Cyclic codes over finite fields form an important and a well-studied class of linear codes. These codes can be effectively encoded and decoded with the help of shift registers due to their nice algebraic structures. As \(\gcd(n,q)=1\), by Maschke's Theorem, the quotient ring \(\mathbb{F}_{q}[x]/\langle x^n-1\rangle \) is semisimple and its minimal ideals are called irreducible cyclic codes of length \(n\) over \(\mathbb{F}_{q}\). Note that if \(f(x)\) is an irreducible factor of \(x^n-1\) over \(\mathbb{F}_q\), then the ideal \(\langle \frac{x^n-1}{f(x)}\rangle\) is a minimal ideal of \(\mathbb{F}_{q}[x]/\langle x^n-1\rangle \), and hence is an irreducible cyclic code of length \(n\) over \(\mathbb{F}_{q}\). In fact, there is a 1-1 correspondence between irreducible factors of \(x^n-1\) over \(\mathbb{F}_{q}\) and irreducible cyclic codes of length \(n\) over \(\mathbb{F}_{q}\). The Hamming weight distribution of a code \(\mathcal{C}\) of length \(n\) over \(\mathbb{F}_{q}\) is defined as the list of numbers \(A_0=1\) \(A_1,A_2,\dots,A_n\), where \(A_i\) denotes the number of codewords in \(\mathcal{C}\) having the Hamming weight \(i\) for \(0 \leq i \leq n\). The Hamming weight distribution of a code is a measure of its error performance relative to several communication channels. However, as pointed out by \textit{C. Ding} [IEEE Trans. Inf. Theory 55, No. 3, 955--960 (2009; Zbl 1367.94388)], the problem of determination of the Hamming weight distributions of an irreducible cyclic code is, in general, a notoriously difficult problem. \textit{B. Schmidt} and \textit{C. White} [Finite Fields Appl. 8, No. 1, 1--17 (2002; Zbl 1023.94016)] provided the following alternative definition of irreducible cyclic codes over finite fields and further classified all irreducible cyclic codes over finite fields with exactly two non-zero Hamming weights. Definition 1. [Schmidt and White, loc. cit.] Let \(k\) be the multiplicative order of \(q\) modulo \(n\), i.e., \(k\) is the smallest positive integer such that \(q^k \equiv 1 ~(\text{mod }n)\). Let \(\omega \) be a primitive \(n\)th root of unity in \(\mathbb{F}_{q^k}\). Then an irreducible cyclic code of length \(n\) over \(\mathbb{F}_{q}\) is the set \[ \left\{\left(Tr_{\mathbb{F}_{q^k}/\mathbb{F}_{q}}(y),Tr_{\mathbb{F}_{q^k}/\mathbb{F}_{q}}(y\omega) , \dots, Tr_{\mathbb{F}_{q^k}/\mathbb{F}_{q}}(y\omega^{n-1})\right): y \in \mathbb{F}_{q^k} \right\},\] where \(Tr_{\mathbb{F}_{q^k}/\mathbb{F}_{q}}\) is the trace function from \(\mathbb{F}_{q^k}\) onto \(\mathbb{F}_{q}\). Later, \textit{G. Vega} [Finite Fields Appl. 33, 1--13 (2015; Zbl 1368.11134)] considered the following more general definition of irreducible cyclic codes over finite fields and characterized all irreducible cyclic codes of length \(n\) and dimension \(k\) over \(\mathbb{F}_q\) having exactly two non-zero Hamming weights. Definition 2. [Vega, loc. cit.] Let \(k\) be the multiplicative order of \(q\) modulo \(n\). For any integer \(a\), let us define \(n=\frac{q^k-1}{\gcd(q^k-1,a)}\). Then the set \[\mathcal{C}(q,k,a)=\left\{\left(Tr_{\mathbb{F}_{q^k}/\mathbb{F}_{q}}(y),Tr_{\mathbb{F}_{q^k}/\mathbb{F}_{q}}(y\omega^a) , \dots, Tr_{\mathbb{F}_{q^k}/\mathbb{F}_{q}}(y\omega^{a(n-1)})\right): y \in \mathbb{F}_{q^k} \right\}\] is called an irreducible cyclic code of length \(n\) and dimension \(k\) over \(\mathbb{F}_q\). The trace description of irreducible cyclic codes over finite fields relates the problem of determination of their Hamming weight distributions with that of the determination of certain character sums, and hence has been found to be useful in computing their Hamming weight distributions in certain special cases. Now recall that \(q\) is a power of the prime number \(p\), and let us suppose that \(\gcd\left( \frac{q^k-1}{n},p\right)=1\). The prime number \(p\) is said to be semiprimitive modulo \( \frac{q^k-1}{n}\) if the integer \(\frac{q^k-1}{n} \) is at least 2 and there exists a positive integer \(j\) satisfying \(p^j\equiv -1\pmod {\frac{q^k-1}{n}} \). The irreducible cyclic code \(\mathcal{C}(p,m,u)\) is said to be semiprimitive if \(u \geq 2\), the prime number \(p\) is semiprimitive modulo \(u\) and \(m\) is the multiplicative order of \(p\) modulo \(u\). Vega [loc. cit.] further generalized this idea and defined semiprimitive irreducible cyclic codes over any finite field as follows: Definition 3. [Vega, loc. cit.] Let \(q\) be a power of the prime number \(p\). An irreducible cyclic code \(\mathcal{C}(q,k,a)\) of length \(n\) and dimension \(k\) over \(\mathbb{F}_q\) is said to be semiprimitive if the integer \(\gcd(\frac{q^k-1}{q-1},a)\) is at least 2 and and the prime number \(p\) is semiprimitive modulo \(\gcd(\frac{q^k-1}{q-1},a)\). The two-weight irreducible cyclic codes that are not semiprimitive are called the exceptional codes. Schmidt and White [loc. cit.] conjectured that there are just 11 exceptional codes. In the case when \(p\) is an odd prime, \(k=2\) and \(2< \frac{q^k-1}{n}< p-1\) with \(\frac{p^k-1}{n} \mid (p-1)\), \textit{A. Rao} and \textit{N. Pinnawala} [IEEE Trans. Inf. Theory 56, No. 6, 2568--2570 (2010; Zbl 1366.94645)] gave a uniform construction of two-weight irreducible cyclic codes \(\mathcal{C}(p,2,\frac{p^k-1}{n})\) without using the McEliece identity [\textit{R. J. McEliece}, in: Combinatorics. Part 1: Theory of designs, finite geometry and coding theory. Proceedings of the Advanced Study Institute on Combinatorics held at Nijenrode Castle, Breukelen, The Netherlands, July 8--20, 1974. Amsterdam: Mathematisch Centrum. 179--196 (1974; Zbl 0309.94022)], which requires \(p-1\) to be a divisor of \(n\). They showed that the irreducible cyclic codes \(\mathcal{C}(p,2,\frac{p^k-1}{n})\) are neither one-weight codes nor semiprimitive codes. They further showed that the smallest value of such a prime number \(p\) is \(19\) when \(p\equiv 3\pmod{4}\) and is \(13\) when \(p\equiv1\pmod{4}\). In this paper, the author presents a simple numerical characterization of all one-weight and semiprimitive two-weight irreducible cyclic codes over finite fields in terms of their lengths. He also provids the following equivalent definition of semiprimitive irreducible cyclic codes in terms of their lengths. Definition 4. Let \(n,k\) and \(u\) be positive integers such that \(\gcd(q,n)=1\), \(k\) be the multiplicative order of \(q\) modulo \(n\), and let \(u=\gcd(\frac{q^k-1}{q-1},\frac{q^k-1}{n})\). Then an irreducible cyclic code of length \(n\) and dimension \(k\) over \(\mathbb{F}_q\) is said to be semiprimitive if \(u\geq 2\) and the prime number \(p\) is semiprimitive modulo \(u\). He also shows that any irreducible cyclic code of length \(n\) and dimension 2 over \(\mathbb{F}_q\) is either a one-weight code or a semiprimitive two-weight code. In more general sense, he shows that all two-weight irreducible cyclic codes of any length \(n\) and dimension \(2\) over \(\mathbb{F}_{q}\) are semiprimitive. Further, in contrast with irreducible cyclic codes of dimension 2 over \(\mathbb{F}_q\), he shows that there exists no semiprimitive two-weight irreducible cyclic code of dimension 3 over any prime field \(\mathbb{F}_{p}\). Besides this, he provides Hamming weight distributions of an infinite family of irreducible cyclic codes over finite fields through the weight distributions of either one-weight irreducible cyclic codes or semiprimitive two-weight irreducible cyclic codes.