Constructive Algorithm of Self-Dual Error-Correcting Codes

1. 11th International Workshop on Algebraic and Combinatorial Coding Theory, June 16-22, Pamporovo, Bulgaria Constructive Algorithm of Self-Dual Error-Correcting Codes Kiyoshi Nagata (DaitoBunka University), Fidel Nemenzo (University of the Pilippines), Hideo Wada (Sophia University)

2. Back Ground of the Research • Discovery of the relationship between non-linear codes over F22 and self-dual linear codes over Z4by A.R. Hammons et al., (IEEE Trans., 1994) • “All self-dual Z4 codes of length 15 or less than known,”J. Fields et al., IEEE Trans., 1998 • “Mass formula for self-dual codes over Z4 and Fq+uFq rings,” P. Gaborit, IEEE Trans., 1996

3. Resent Works • “On the number of distinct self-dual codes over Z9,”J.M. P. Balmaceda, R. A. L. Betty and F. R. Nemenzo, Matimyas Matematika 26, 2003, pp.9-7 • “Mass formula for self-dual codes over Zp2,”J.M. P. Balmaceda, R. A. L. Betty and F. R. Nemenzo, (to appear in Discrete Mathematics) • “The number of self-dual codes over Zp3,”K. Nagata, F. Nemenzo, and H. Wada, (to appear in Designs, Codes and Cryptography) • “Mass Formula for Self-Orthogonal Codes over Zp2,”R. A. Betty and A. Munemasa,(e-print in http://jp.arxiv.org/abs/0805.2205)

4. Situation • Assumptions • p: an odd prime number • s: any integer greater than 4 • Definitions • Zps= Z/psZ the integer ring modulo ps • C : a linear code over Zps with k-generators • C⊥={u: a code word s.t. uC =0} “the dual of C “ • C is self-orthogonal ⇔ C ⊆C⊥ • C is self-dual ⇔ C = C⊥

5. Result • Recursive Algorithm for Construction of self-dual codes over Zps

6. Basic Representation of a Linear Code and Its Dual After some permutation of columns and some row operations Using [ Ti*] t = [ Ti]-1 with

7. Basic Proposition and Lemma

8. From C ‘ over Zps-2 to C over Zps From self-dual code over Zps-2 To the the self-code over Zps (Some kind of modification)

9. Condition for Ts (Ass) , Ti’s (Ais(1)’s), and T1 (A1s-1(1), A1s(1), and A1s(2)) Uniquely determined from C‘! Uniquely determined from C‘! Uniquely determined from C‘, A1s-1(1) and A1s(1) ! Condition for A1s-1(1) and A1s(1) with

10. Condition for A1s(1) and A1s(2) • Case of odd prime p Put the right-hand side ,and put

11. Put the right-hand side ,and put

12. (Pre-) Mass Formula

13. Condition for A1s(1) and A1s(2) • Case of p=2 Necessary additional condition Put ,and put and for any xii.

14. Put the right-hand side ,and put

15. Example for p=2, s=4, n=6

16. and Then

17. Future Works! • Complete determination of the Number of self-dual codes over Zps for any prime p and for any integer s>0 • Find some good non-linear code over Fps using an extended Gray map

18. Thank You!