On Self-dual Codes Darwin Villar July 2018 Introduction Codes and - PowerPoint PPT Presentation

On Self-dual Codes Darwin Villar July 2018

Introduction Codes and monomial groups Generalized Code [52,26,15] Generalization Open Questions Introduction Let C be a self-dual [ n , k , d ]- code over F q . Type I C is 2 -divisible or even and q = 2 Type II C is 4 -divisible or doubly even and q = 2 Type III C is 3 -divisible and q = 3 Type IV C is 2 -divisible and q = 4 In 1973 C.L. Mallows and N.J.A. Sloane proved that the minimum distance d of a self-dual [ n , k , d ]-code satisfies � n � Type I + 2 d ≤ 2 � n � 8 Type II + 4 , if n �≡ 22 mod 24 d ≤ 4 � n � 24 d ≤ 4 + 6 , if n ≡ 22 mod 24 � n � 24 Type III + 3 d ≤ 3 � n � 12 Type IV + 2 d ≤ 2 6 Codes reaching the bound are called Extremal .

Introduction Codes and monomial groups Generalized Code [52,26,15] Generalization Open Questions The known extremal ternary codes of length 12 n . Extremal Partial Length n S ( n 2 − 1 ) XQR ( n − 1 ) Classification ∗ distance 12 6 6 � 24 9 9 9 � 36 12 - 12 o ( σ ) ≥ 5 48 15 15 15 o ( σ ) ≥ 5 60 18 18 18 o ( σ ) ≥ 11 72 - 18 21 No extremal ∗ σ ∈ Aut( C ) of prime order.

Introduction Codes and monomial groups Generalized Code [52,26,15] Generalization Open Questions Generalized Code Minimum distance of the Pless codes computed with Magma . 5 11 17 23 29 41 47 p 12 24 36 48 60 84 96 2 ( p + 1 ) 6 9 12 15 18 21 24 d ( P 3 ( p )) Aut( P 3 ( p )) 2 . M 12 G ( 11 ) . 2 G ( 17 ) . 2 G ( 23 ) . 2 G ( 29 ) . 2 ≥ G ( 41 ) ≥ G ( 47 ) For q = 5 , 7 , and 11 we computed d ( P q ( p )) with Magma : ( p , q ) ( 11 , 5 )( 19 , 5 )( 29 , 5 )( 31 , 5 ) ( 3 , 7 )( 5 , 7 )( 13 , 7 ) 2 ( p + 1 ) 12 40 60 64 8 12 28 d ( P q ( p )) 9 13 18 18 4 6 10 ( p , q ) ( 17 , 7 )( 19 , 7 ) ( 7 , 11 )( 13 , 11 )( 17 , 11 )( 19 , 11 ) 2 ( p + 1 ) 36 40 16 28 36 40 d ( P q ( p )) 12 13 7 10 12 13

Introduction Codes and monomial groups Generalized Code [52,26,15] Generalization Open Questions The new series of Codes Minimum distance of V 3 ( p ) computed with Magma : p 5 13 29 37 53 2 ( p + 1 ) 12 28 60 76 108 d ( V 3 ( p )) 6 9 18 18 24 Aut( V 3 ( p )) 2 . M 12 SL 2 ( 13 ) SL 2 ( 29 ) ≥ SL 2 ( 37 ) ≥ SL 2 ( 53 ) For q = 5 , 7 , and 11 and small lengths we computed d ( V q ( p )) with Magma : ( p , q ) ( 13 , 5 ) ( 29 , 5 ) ( 5 , 7 ) ( 13 , 7 ) ( 5 , 11 ) ( 13 , 11 ) 2 ( p + 1 ) 28 60 12 28 12 28 d ( V q ( p )) 10 16 6 9 7 11

Introduction Codes and monomial groups Generalized Code [52,26,15] Generalization Open Questions Thanks for your attention