A Study on Minimal Cyclic Codes of Length pnq

To analyze idempotents, the following results are necessary: Lemma 9: If β is a primitive pkth root of unity for an odd prime p and k a positive integer,, then where is a primitive root mod pk. Proof: See Bakshi G. K. and Raka Madhu [2, Lemma 4]. Let α be a fixed primitive root of unity in some extension field of . For , , define Lemma 10. For each , Proof: For any k and i, (mod pnq) if and only if if and only if . Therefore, Where is a primitive root of unity. Therefore, is a reduced residue system mod , the sum on the R.H.S. is As , the first three geometric series are zero. If , i.e. if , the sum of the last series also vanishes. If , the last series is . Thus, for and 0 for . Lemma 11: For each k, and , Proof: For and , so that the required sum (using Lemma 9) is Where is a primitive qth root of unity, if . Therefore, if and only if (mod q), Only if s ≡ r mod q is accessible. The sum is then equivalent to Lemma 9. From, The lemma sum is the same . The lemma proves this. Lemma 12: For , Proof: Let r be either q or ak for some k, . The sum expected is the same as This completes the lemma proof. Lemma 13: For , Proof: For , as , the required sum is , where . For the sum is Where is a primitive root of unity for and if then β = . Therefore the sum is the same, if i + j n. If , The sum becomes then by Lemma 9 The lemma proves this codes of length 2pn, Finite Fields Appl. 5, pp. 177–187. [2] G.K. Bakshi & Madhu Raka (2003). Minimal cyclic codes of length pnq, Finite Fields Appl. 9, pp. pp. 432–448. [3] F.J. MacWilliams & N.J.A. Sloane (1977). Theory of Error-Correcting Codes, North-Holland, Amsterdam, 1977. [4] Vera Pless : Introduction to the Theory of Error-Correcting Codes, Wiley–Intersci. Ser. Discrete Math. Optim. [5] M. Pruthi & S.K. Arora (1997). Minimal cyclic codes of prime power length, Finite Fields Appl. 3, pp. 99–113. [6] M. Pruthi (2001). Cyclic codes of length 2m, Proc. Indian Acad. Sci. Math. Sci. 111, pp. 371–379. [7] A. Sharma, G.K. Bakshi, V.C. Dumir, M. Raka (2008). Irreducible cyclic codes of length 2n, Ars Combin. 86, pp. 133–146. [8] Ranjeet Singh, Manju Pruthi (2011). Primitive idempotents of irreducible quadratic residue cyclic codes of length pnqm, Int. J. Algebra 5, pp. 285–294. [9] S.K. Arora and M. Pruthi (1999). “Minimal Cyclic Codes Length pn ” Finite Field and their Applications, 5, pp. 177-187. [10] Vera Pless (1988). “Introduction to the Theory of Error-Correcting Codes” WileyIntersci. Ser. Discrete Math. Optim.,

Research Scholar, Department of Mathematics, Sai Nath University, Ranchi, Jharkhand