ISSN 2148-838Xhttps://doi.org/10.13069/jacodesmath.1000837 J. Algebra Comb. Discrete Appl. 8(3) • 167–195 Received: 21 September 2020 Accepted: 08 March 2021 Journal of Algebra Combinatorics Discrete Structures and Applications Minimum distance and idempotent generators of minimal cyclic codes of length p1 α1p2 α2p3 α3 Research Article Pankaj Kumar, Pinki Devi Abstract: Let p1, p2, p3, q be distinct primes and m = p1α1p2α2p3α3 . In this paper, it is shown that the explicit expressions of primitive idempotents in the semi-simple ring Rm = Fq[x]/(xm − 1) are the trace function of explicit expressions of primitive idempotents from R p αi i . The minimal polynomials, generating polynomials and minimum distances of minimal cyclic codes of length m over Fq are also discussed. All the results obtained in [1], [3], [4], [5], [11] and [14] are simple corollaries to the results obtained in the paper. 2010 MSC: 94B05, 94B15 Keywords: Cyclotomic cosets, Primitive idempotents, Cyclic codes, Trace function 1. Introduction Let p1,p2,p3 and q be distinct primes and m = p1α1p2α2p3α3.Then Rm = Fq[x]/(xm −1) is a semi- simple ring. Every cyclic code C of length m over Fq is a direct sum of minimal cyclic codes in Rm. Therefore, we need to compute the minimal cyclic codes of length m over Fq. A minimal cyclic code always has a primitive idempotent generator [12, Theorem 1, Chapter 8], therefore, the study of minimal cyclic codes need the computation of explicit expressions of primitive idempotents in Rm. For m = pn, Pruthi and Arora [13] obtained all the minimal cyclic codes of length m in Rpn = Fq[x]/(xp n −1); q is a primitive root modulo pn. In [7], Chen et al. studied the minimal cyclic codes of length lm over Fq, where l is a prime divisor of q − 1 and m is a positive integer. Many authors have worked to compute the primitive central idempotents in semi-simple group algebra. In [2], Bakshi et al. have computed primitive central idempotents in semi-simple group algebra FG; F is finite fields and G is arbitrary meta-cyclic group. In another paper [9], they developed an algorithm to compute the primitive central idempotents in a semi-simple group algebra. Broche and Rio [6] developed a method to compute the primitive central idempotents and the wedderburn decomposition of a semi simple finite group algebra of Pankaj Kumar, Pinki Devi (Corresponding Author); Department of Mathematics, Guru Jambheshwar University of Science and Technology, Hisar 125001, India (email: joshi78023@yahoo.com, pinkinarwal123@gmail.com). 167 https://orcid.org/0000-0002-3371-1875 https://orcid.org/0000-0002-3245-8863 P. Kumar, P. Devi / J. Algebra Comb. Discrete Appl. 8(3) (2021) 167–195 an abelian-by-supersolvable group G. Ferraz and Milies [8] discussed a simple method of computing the idempotent generator of minimal abelain codes and discussed all the results as obtained in [1]. Sharma et al. [15] gave an algorithm to compute all the primitive idempotents over Fq in Rpn by choosing fpn−1 as the order of q modulo pn. Since m = p1α1p2α2p3α3, therefore it is natural to ask that “Can one use the explicit expressions of primitive idempotents from Rpαi i to compute the explicit expressions of primitive idempotents in Rm?” In this regard Bakshi et al. [4] used trace function to compute the explicit expressions of primitive idempotents in Rm; m = m1m2 with gcd ( Om1(q),Om2(q) ) = 1 or d, where Omi(q) denotes the order of q modulo mi. Kumar and Arora [10] defined a λ-mapping to compute the explicit expressions of primitive idempotents in Rm; m = p α1 1 p α2 2 ...p αr r with the help of primitive idempotents from Rpαi i in a single step under the following conditions: (i) For each i, q is a primitive root modulo pαii , (ii) At most one prime factor of m is of the form 4k + 1 and the other prime factors of m are of the form 4k + 3, (iii) gcd ( φ(pαii ),φ(p αj j ) ) = 2, 1 ≤ i 6= j ≤ r. In this paper, we choose m = p1α1p2α2p3α3, where pi’s are distinct primes and, for each i,1 ≤ i ≤ 3, Opαi i (q) = φ(p αi i ) ri and obtain the primitive idempotents and minimum distances of minimal cyclic codes in Rm over the finite field Fq. This paper is organized as follows: In Section 2, we use λ- mapping to obtain the q-cyclotomic cosets and in Theorem 2.5, we count the number of distinct q-cyclotomic cosets modulo m. In Section 3, the primitive idempotents are obtained. In Theorem 3.5, we use λ-mapping and in Theorem 3.6 we use Trace function to compute the primitive idempotents in Rm with the help of primitive idempotents from Rpαi i . The minimal polynomials and the generating polynomials of some cyclic codes of length m are obtained in Section 4. In Section 5, we give a lower bound on minimum distances of cyclic codes of length m. We include examples in the last section. In Example 6.1, we choose m = 715 and count the total number of 3-cyclotomic cosets modulo 715. Then we obtain the explicit expression of θ7151 (x) in R715 by λ product of polynomials and by Trace function (The explicit expressions of all primitive idempotents are shown in Appendix 1). In Example 6.2, the minimal cyclic codes of length 30 are discussed in R30 = F7[x]/(x30 −1). All the results obtained in [1], [3], [4], [5], [11] and [14] are simple corollaries to the results obtained in the paper. 2. Cyclotomic cosets modulo m In this section we use λ-mapping [10, Definition(2.2)] to compute the q-cyclotomic cosets modulo m with the help of qvi-cyclotomic cosets modulo pαii . Then we count the number of distinct q-cyclotomic cosets modulo m. Throughout this paper p1,p2,p3 and q are distinct primes, m = p1α1p2α2p3α3, for 1 ≤ i ≤ 3, Opαi i (q) = φ(p αi i ) ri , mi = mpαi i , A = {0,1,2, ...,m − 1} and Ai = {0,1, ...,pαii − 1}, δ is a fixed primitive mth root of unity in some extension of Fqt and δi = δmi is a fixed primitive piαith root of unity in the same extension of Fqt. We denote the q-cyclotomic coset modulo m containing s; 0 ≤ s ≤ m−1 by C m(q) s = {s,sq,sq2, ...,sqr−1}, where r is the smallest positive integer such that sqr ≡ s(mod m) and, for a divisor i of r, Cm(q i) s = {s,sqi,sq2i, ...,sqi( r i −1)} denote the qi-cyclotomic coset modulo m containing s. Definition 2.1. [λ-mapping] The mapping λ from A1 ×A2 ×A3 −→ A defined by λ(b1,b2,b3) = b1m1 + b2m2 + b3m3(mod m), where bi ∈ Ai is called the λ-mapping. It is a one-one and onto mapping. Lemma 2.2. Let v1,v2 and v3 be positive integers. If l = lcm(v1,v2,v3), l1 = lcm(v2,v3), l2 = lcm(v1,v3) and l3 = lcm(v1,v2) then, for 1 ≤ i ≤ 3 (i) li divides l. (ii) l divides vili and, if vili = uil, where ui is some positive integer, then l1l2l3 = l2lcm(u1,u2,u3). 168 P. Kumar, P. Devi / J. Algebra Comb. Discrete Appl. 8(3) (2021) 167–195 Theorem 2.3. Let q be an odd prime with gcd(q,m) = 1. Further, let Om(q) = d, Omi(q) = di, vi = O p αi i (q)di d . Then C m(q) 1 = t−1⋃ j=0 λ ( C p α1 1 (q v1 ) a1qj ×Cp α2 2 (q v2 ) a2qj ×Cp α3 3 (q v3 ) a3qj ) , where t = lcm(v1,v2,v3). Proof. By Definition 2.1, for 1 ∈ A , there exists (a1,a2,a3) ∈ A1 ×A2 ×A3 such that λ(a1,a2,a3) = 1. (1) As Om(q) = d, therefore, multiplying (1) by qk; 0 ≤ k ≤ d−1, we get λ(a1,a2,a3) = 1 λ(a1q,a2q,a3q) = q λ(a1q 2,a2q 2,a3q 2) = q2 ... λ(a1q d−1,a2q d−1,a3q d−1) = qd−1. The left hand side of above equations has three columns namely B1 = {a1,a1q,a1q2, . . . ,a1qd−1}, B2 = {a2,a2q,a2q2, . . . ,a2qd−1}, B3 = {a3,a3q,a3q2, . . . ,a3qd−1}. As Om1(q) = d1 and v1 = O p α1 1 (q)d1 d , therefore, the subset Cp α1 1 (q v1 ) a1 of B1 will repeat with the set {(a2,a3),(a2q,a3q),(a2q2,a3q2), . . . ,(a2qd1−1,a3qd1−1)}. Similarly, the subset Cp α2 2 (q v2 ) a2 of B2 will repeat with the set {(a1,a3),(a1q,a3q),(a1q2,a3q2), . . . ,(a1qd2−1,a3qd2−1)} and Cp α3 3 (q v3 ) a3 of B3 will repeat with the set {(a1,a2),(a1q,a2q),(a1q2,a2q2), . . . ,(a1qd3−1,a2qd3−1)}. Hence C p α1 1 (q v1 ) a1 ×C p α2 2 (q v2 ) a2 ×C p α3 3 (q v3 ) a3 ⊂{(a1,a2,a3),(a1q,a2q,a3q), . . . ,(a1qd−1,a2qd−1,a3qd−1)}. 169 P. Kumar, P. Devi / J. Algebra Comb. Discrete Appl. 8(3) (2021) 167–195 Similarly, we get C p α1 1 (q v1 ) a1q ×C p α2 2 (q v2 ) a2q ×C p α3 3 (q v3 ) a3q ⊂{(a1,a2,a3),(a1q,a2q,a3q), . . . ,(a1qd−1,a2qd−1,a3qd−1)}. As the set {(a1,a2,a3),(a1q,a2q,a3q), . . . ,(a1qd−1,a2qd−1,a3qd−1)} is finite, therefore, there exists a positive integer t; t = lcm(v1,v2,v3) such that t−1⋃ j=0 ( C p α1 1 (q v1 ) a1qj ×Cp α2 2 (q v2 ) a2qj ×Cp α3 3 (q v3 ) a3qj ) ⊂{(a1,a2,a3),(a1q,a2q,a3q), ...,(a1qd−1,a2qd−1,a3qd−1)}. Since C p αi i (qvi) aiqj contains d di elements, therefore, t−1⋃ j=0 ( C p α1 1 (q v1 ) a1qj ×Cp α2 2 (q v2 ) a2qj ×Cp α3 3 (q v3 ) a3qj ) contains td 3 d1d2d3 elements. (By Lemma 2.2) td 3 d1d2d3 = d, therefore, t−1⋃ j=0 ( C p α1 1 (q v1 ) a1qj ×Cp α2 2 (q v2 ) a2qj ×Cp α3 3 (q v3 ) a3qj ) = {(a1,a2,a3),(a1q,a2q,a3q), . . . ,(a1qd−1,a2qd−1,a3qd−1)}. Equivalently, t−1⋃ j=0 λ ( C p α1 1 (q v1 ) a1qj ×Cp α2 2 (q v2 ) a2qj ×Cp α3 3 (q v3 ) a3qj ) = λ{(a1,a2,a3),(a1q,a2q,a3q), . . . ,(a1qd−1,a2qd−1,a3qd−1)} = C m(q) 1 . Corollary 2.4. Let s ∈ A. If ais ≡ bip βi i (mod p αi i ) then Cm(q)s = t−1⋃ j=0 λ ( C p α1 1 (q v1 ) b1p β1 1 q j ×Cp α2 2 (q v2 ) b2p β2 2 q j ×Cp α3 3 (q v3 ) b3p β3 3 q j ) . Proof. By Theorem 2.3, C m(q) 1 = t−1⋃ j=0 λ ( C p α1 1 (q v1 ) a1qj ×Cp α2 2 (q v2 ) a2qj ×Cp α3 3 (q v3 ) a3qj ) , therefore, for s ∈ A, we have Cm(q)s = t−1⋃ j=0 λ ( C p α1 1 (q v1 ) a1sqj ×Cp α2 2 (q v2 ) a2sqj ×Cp α3 3 (q v3 ) a3sqj ) . 170 P. Kumar, P. Devi / J. Algebra Comb. Discrete Appl. 8(3) (2021) 167–195 If ais ≡ bip βi i (mod p αi i ) then clearly Cm(q)s = t−1⋃ j=0 λ ( C p α1 1 (q v1 ) b1p β1 1 q j ×Cp α2 2 (q v2 ) b2p β2 2 q j ×Cp α3 3 (q v3 ) b3p β3 3 q j ) . Theorem 2.5. Let q be an odd prime with gcd(q,m) = 1. Further, let Opαi i (q) = φ(p αi i ) ri , Om(q) = d, Omi(q) = di, vi = O p αi i (q)di d and lcm(v1,v2,v3) = t. Then, the number of distinct q-cyclotomic cosets modulo m is ∏3 i=1 riαivi t + ∑ 1≤i