Good Integers and Applications in Coding Theory. Silpakorn University

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1 Good Integers and in Coding Theory Somphong Jitman Silpakorn University March 1,

2 God made the integers, all else is the work of man. L. Kronecker 1 Good Integers 2

3 Good Integers P. Moree (1997) For given co-prime positive integers a and b, a positive integer l is said to be good (w.r.t. a and b) if it is a divisor of a k + b k for some positive integer k. Example Let a = 2 and b = 3. Then the following statements hold. 1 and 5 are good since they are divisors of 5 = = a + b. 7 is good since it is a divisor of 35 = = a 3 + b 3. 2 is not a good integer since a k + b k = 2 k + 3 k is odd for every positive integer k.

4 Good Integers P. Moree (1997) For given co-prime positive integers a and b, a positive integer l is said to be good (w.r.t. a and b) if it is a divisor of a k + b k for some positive integer k. Problems Is 2 10 good w.r.t. 3 and 5? Is 28 good w.r.t. 3 and 5? In general, how to determine the goodness of (large) integers?

5 D. Knee and H. D. Goldman (1969) (before 1997) Good integers l with a = q (q is a prime power) and b = 1 are studied and applied in constructing BCH codes with good design distances. G. Skersys (2003) Good integers l with a = q (q is a prime power) and b = 1 are applied in determining the average dimensions of hull of cyclic codes over finite fields. Y. Jia, S. Ling, and C, Xing (2011) Good integers l with a = 2 l and b = 1 are applied in determining the number of Euclidean self-dual cyclic codes over finite fields.

6 S. Jitman, S. Ling, H. Liu, X. Xie (2013) Good integers l with a = 2 m and b = 1 are applied in determining the number of Euclidean self-dual abelian codes over finite fields. E. Sangwisut, S. Jitman, S. Ling, and P. Udomkavanich (2015) Good integers l with a = q (q is a prime power) and b = 1 are applied in determining the number of Euclidean and Hermitian complementary dual cyclic codes over finite fields.

7 Lemma Let a and b be nonzero coprime integers and let d be a positive integer. If l G (a,b), then gcd(a, l) = 1 = gcd(b, l). If ab is even, then there exists only an odd good integer. If ab id odd, then there exist both odd and even good integers. P. Moree (1997) Let d > 1 be an odd integer. Then l G (a,b) if and only if there exists s 1 such that 2 s ord p (ab 1 ) for every prime p d.

10 Problems Is 2 10 good w.r.t. 3 and 5? [2 β with β 2 is good if and only if 2 β (a + b).] 2 10 is not good w.r.t. 3 and 5 but 2 2 and 2 3 are good w.r.t. 3 and 5. Is 28 good w.r.t. 3 and 5? [2 β d (with β 2 and d is odd) is good if and only if 2 β is good and 2 ord d (ab 1 ).] Since mod 7 and ord 7 (3 5 1 ) = ord 7 (2) = 3, 28 = is not good w.r.t 3 and 5. In general, how to determine the goodness of (large) integers?

11 Oddly-Good and Evenly-Good Integers Definition For co-prime positive integers a and b, an integer l 1 is said to be oddly good w.r.t. a and b if l ( a k + b k) for some odd k N, evenly good w.r.t. a and b if l ( a k + b k) for some even k N, good if it is oddly good or evenly good bad, otherwise. Example Let a = 8 and b = 1. Then the following statements holds. 3 is oddly good since 3 ( ). 5 and 13 are evenly good since they are divisors of 65 = is bad since 8 k + 1 k is odd for every positive integer k.

12 1 is always good. Since 1 (a + b) and 1 ( a 2 + b 2), 1 is both oddly-good and evenly-good. 2 is good if and only if ab is odd. In this case, a + b and a 2 + b 2 are even, and hence, 2 (a + b) and 2 (a 2 + b 2 ) which imply that 2 is both oddly-good and evenly-good. Proposition Let a, b and l > 2 be pairwise coprime nonzero integers. If l G (a,b), then either l OG (a,b) or l EG (a,b), but not both.

15 of Good Integers in Coding Theory

16 Linear Codes Good Integers For a prime power q, denote by F q the finite field of q elements. (For instance, consider the prime field F p = Z p.) A set C F n q is called a linear code of length n over F q if C is a subspace of the F q -vector space F n q Example C = {0000, 1010, 0101, 1111} is a linear code of length 4 over F 2. D = {00000, 11111} is a linear code of length 5 over F 2.

17 The Euclidean dual of C of length n over F q is defined to be C E = {u F n q u, c E = 0 for all c C}, where u, v E = n i=1 u iv i. C is Euclidean self-dual if C = C E. C is Euclidean complementary dual if C C E = {0}. Example C = {0000, 1010, 0101, 1111} C E = C. D = {00000, 11111} D E = 00000, 11000, 01100, 00110, and D D E = {00000}.

18 Definition A linear code C of legth n over F q is said to be cyclic if (c n 1, c 0, c 1,..., c n 2 ) C whenever (c 0, c 1,..., c n 1 ) C. Example C = {0000, 1010, 0101, 1111} is cyclic over F 2. D = {00000, 11111} is cyclic over F 2. Shift Registers and Cyclic Codes

19 Let π : F n q F q [x]/ x n 1 be an F q -linear isom. given by π((v 0, v 1,..., v n 1 )) = v 0 + v 1 x + + v n 1 x n 1. Theorem Let C be a linear code of length n over F q. Then C is cyclic if and only if π(c) is an ideal in the principal ideal ring F q [x]/ x n 1. In this case, π(c) is uniquely generated by a monic divisor G(x) of x n 1 of minimal degree in π(c).

20 Factorization of x n 1 over F q Assume that the characteristic of F q is p. Then n = p ν n for some ν 0 and p n. For a, b F q, we have (a + b) p = a p + b p. Hence, x n 1 = x pν n 1 pν = (x n 1) pν.

21 For 0 j < n, let C q (j) = {jq i modn i Z} be the q-cyclotomic coset of j modulo n. {0, 1,..., n 1} = r C q (j i ) is disjoint for some r. x n 1 = r m i (x) i=1 i=1 where m i (x) = (x ω l ) is the minimal polynomial of ω j i over F q. l C q(j i ) m i (x) := l C q (j i ) (x ω l ). m i (x) = m i (x) if and only if C q(j i ) = C q ( j i ) {0, 1,..., n 1} = s C q (a i ) t C q (b j ) C q ( b j ) i=1 x n 1 = s f i (x) i=1 t j=1 j=1 g j (x)g j (x).

22 Euclidean Complementary Dual Cyclic Codes s t x n 1 = (x n 1) pν = f i (x) pν g j (x) pν gj (x) pν i=1 j=1 Proposition A cyclic code C of length n over F q with the generator polynomial G(x) is Euclidean complementary dual if and only if G(x) = s f i (x) α i t (g j (x)gj (x))β j, where α i, β j {0, p ν }. i=1 j=1 The number of Euclidean complementary dual cyclic codes of length n over F q is 2 s+t.

23 Euclidean Self-Dual Cyclic Codes Lemma There exists a Euclidean self-dual cylcic code of length n over F q if and only if q and n are even. x n 1 = (x n 1) 2ν = Proposition s f i (x) 2ν i=1 t g j (x) 2ν gj (x) 2ν, ν > 0 A cyclic code C of length n = 2 ν n over F 2 m with the generator polynomial G(x) is Euclidean self-dual if and only if G(x) = s t f i (x) 2ν 1 g j (x) βj gj β j, where 0 β (x)2ν j 2 ν. i=1 j=1 The number of Euclidean self-dual cyclic codes of length n = 2 ν n over F 2 m is (2 ν + 1) t. j=1

24 s t x n 1 = (x n 1) pν = f i (x) pν g j (x) pν gj (x)pν i=1 j=1 C q (a i ) = C q ( a i ) C q (b j ) C q ( b j ) Lemma Let a, b {0, 1, 2,..., n 1}. Then the following statements hold. C q(a) = C q( a) if and only if a G (q,1). C q (a) = ord ord(a) (q). If C q (a) = C q (b), then C q (a) = C q ( a) if and only if C q (b) = C q ( b). Proposition s = d n d G (q,1) ϕ(d) ord d (q) and t = 1 2 d n d/ G (q,1) ϕ(d) ord d (q).

25 Hermitian Case over F q 2 s = t = 1 2 d n d OG (q,1) d n d/ OG (q,1) ϕ(d) ord d (q 2 ). ϕ(d) ord d (q 2 ).

26 Summary Good Integers Good BCH codes D. Knee, H. D. Goldman, Quasi-self-reciprocal polynomials and potentially large minimum distance BCH codes, IEEE Trans. Inform. Theory 15 (1969) The average dimension of the Euclidean hull of cyclic codes G. Skersys, The average dimension of the hull of cyclic codes, Discrete Applied Mathematics 128 (2003) The enumeration of Euclidean self-dual cyclic codes Y. Jia, S. Ling, and C. Xing, On self-dual cyclic codes over finite fields, IEEE Trans. Inform. Theory 57 (2011) The enumeration of Euclidean self-dual abelian codes S, Jitman, S. Ling, H. Liu, and X. Xie, Abelian codes in principal ideal group algebras, IEEE Trans. Inform. Theory 59 (2013) The enumeration of Hermitian self-dual abelian codes S. Jitman, S. Ling, P. Solé, Hermitian self-dual Abelian codes, IEEE Trans. Inform. Theory 60 (2014)

27 Summary (Cont ) Good Integers The enumeration of Euclidean/Hermitian Complementary dual abelian codes S. Jitman, E. Sangwisut, The average dimension of the Hermitian hull of cyclic codes over finite fields of square order, AIP Conference Proceedings bf 1775 (2016) The average dimension of the Hermitian hull of cyclic codes S. Jitman, E. Sangwisut, The average dimension of the Hermitian hull of cyclic codes over finite fields of Square Order, AIP Proceeding of the International Conference on Mathematics, Engineering and Industrial (IC0MEIA2016) 1775, Article ID (2016). The average dimension of the Hermitian hull of constacyclic codes S. Jitman, E. Sangwisut, The average dimension of the Hermitian hull of constayclic codes over finite fields, The average dimension of the Euclidean/Hermitian hull of abelian codes S. Jitman, Good integers and applications in coding theory,

28 Somphong Jitman - SJitman@Gmail.com

Repeated-Root Self-Dual Negacyclic Codes over Finite Fields

Journal of Mathematical Research with Applications May, 2016, Vol. 36, No. 3, pp. 275 284 DOI:10.3770/j.issn:2095-2651.2016.03.004 Http://jmre.dlut.edu.cn Repeated-Root Self-Dual Negacyclic Codes over