A classification of self-dual codes with a rank 3 automorphism group of almost simple type Bernardo Rodrigues School of Mathematics, Statistics and Computer Science University of KwaZulu-Natal Durban, South Africa Groups St Andrews 2017 University of Birmingham, August 2017 Bernardo Rodrigues, rodrigues@ukzn.ac.za (University of KwaZulu-Natal) self-dual codes and rank 3 groups 06 August 2017 1 / 30
The problem and motivation Problem 1 Given a permutation group G of degree n acting rank 3 on a set Ω determine all self-dual codes C of length n on which G acts transitively on the code coordinates. The rank of a permutation group G transitive on a set Ω is the number of orbits of G ω , ω a point of Ω , in Ω . A transitive group G has rank 2 on the set Ω if and only if G is 2-transitive on Ω . G has rank 3 if and only if for every point ω in Ω , G ω has two orbits besides G ω . Rank 3 groups can be either primitive or imprimitive. Bernardo Rodrigues, rodrigues@ukzn.ac.za (University of KwaZulu-Natal) self-dual codes and rank 3 groups 06 August 2017 2 / 30
Self-Dual Codes We consider binary self-dual codes invariant under permutation groups A binary linear code C is a subspace of F n 2 The dual code C ⊥ is defined as : C ⊥ := { v |� u , v � = 0 for all u ∈ C } The Hamming weight of a codeword c ∈ C is wt ( c ) := |{ i | c i � = 0 }| The minimum distance d ( C ) = d of a code C is the smallest of the distances between distinct codewords; i.e. d ( C ) = min { d ( v , w ) | v , w ∈ C , v � = w } . A code C denoted [ n , k , d ] q is said to be of length n , dimension k and minimum distance d over the field of q -elements. Bernardo Rodrigues, rodrigues@ukzn.ac.za (University of KwaZulu-Natal) self-dual codes and rank 3 groups 06 August 2017 3 / 30
C can detect up to d − 1 errors or correct up to ⌊ ( d − 1 ) / 2 ⌋ errors. C is self-orthogonal if C ⊂ C ⊥ If C = C ⊥ the code is self-dual If a code has all its weights divisible by 4 then it is called doubly even(Type II) The length n of a doubly even code is a multiple of 8; For a self-dual code C we have dim ( C ) = n 2 and all codewords have even weight For a self-dual code: � 4 ⌊ n 24 ⌋ + 4 , if n �≡ 22 (mod 24 ) d ≤ 4 ⌊ n 24 ⌋ + 6 , if n ≡ 22 (mod 24 ) If “ = ” then the code is called extremal Bernardo Rodrigues, rodrigues@ukzn.ac.za (University of KwaZulu-Natal) self-dual codes and rank 3 groups 06 August 2017 4 / 30
Module Structure Let G ≤ Aut ( C ) For x ∈ F qn and a permutation σ ∈ S n we set σ x = ( x σ − 1 ( 1 ) , x σ − 1 ( 2 ) , . . . x σ − 1 ( n ) ) . (1) Aut ( C ) = { σ ∈ S n | σ x ∈ C for all x ∈ C } C ≤ F n q as F G -modules ( � σ x , σ y � = � x , y � , for x , y ∈ F n q , σ ∈ G C ⊥ is also a F G -module Aut ( C ) = Aut ( C ⊥ ) C ∗ = Hom F ( C , F ) becomes a F G -module via σ ( f )( c ) = f ( σ − 1 ( c )) q / C ⊥ ∼ = C ∗ as F G -modules F n Bernardo Rodrigues, rodrigues@ukzn.ac.za (University of KwaZulu-Natal) self-dual codes and rank 3 groups 06 August 2017 5 / 30
What is known ... thus far? Example 1 (Extended cyclic code) σ = ( 1 2 3 4 5 6 7 ) - cyclic shift, ( 8 ) is fixed. 0 1 1 0 1 0 0 1 0 0 1 1 0 1 0 1 0 0 1 1 0 1 0 1 0 0 0 1 1 0 1 1 σ h 8 := �→ 0 0 0 1 1 0 1 1 1 0 0 0 1 1 0 1 1 0 0 0 1 1 0 1 0 1 0 0 0 1 1 1 TABLE 1: Known extremal self-dual doubly even codes Length 8 24 32 40 48 72 80 ≥ 3928 d ( C ) 4 8 8 8 12 16 16 extremal h 8 G 24 5 16,470 QR 48 ? ≥ 4 0 Bernardo Rodrigues, rodrigues@ukzn.ac.za (University of KwaZulu-Natal) self-dual codes and rank 3 groups 06 August 2017 6 / 30
Automorphism Group Aut ( h 8 ) = 2 3 : L 3 ( 2 ) Aut ( G 24 ) = M 24 Length 32: L 2 ( 31 ); 2 5 : L 5 ( 2 ); 2 8 : S 8 , ( 2 8 : L 2 ( 7 )): 2 , 2 5 : S 6 . Length 40: 10,400 extremal codes with Aut = 1 . Aut ( QR 48 ) = L 2 ( 47 ) . Sloane (1973): Is there a [ 72 , 36 , 16 ] self-dual code? Still open Extremal codes only known for n = 8 , 16 , 24 , 32 , 40 , 48 , 56 , 64 , 80 , 88 , 104 , 112 , 136 ? 136 ≤ . . . ≤ 3928 Bernardo Rodrigues, rodrigues@ukzn.ac.za (University of KwaZulu-Natal) self-dual codes and rank 3 groups 06 August 2017 7 / 30
2-Transitive Automorphism Groups Question 1 Given a permutation group G of degree n acting rank 2 on a set Ω determine all self-dual extremal codes C of length n on which G acts transitively on the code coordinates. It is well-known that every 2-transitive group is primitive. By using CFSG, all finite 2-transitive groups are known. G = Aut ( C ) is 2-transitive Use the structure of G 1 ⋆ The socle of G is simple or elementary abelian ⋆ Degree of G = length of C ≤ 3928 ⋆ ⇒ Only few possibilities for G Find all FG -modules of dim n 2 2 Find modules that are self-dual as codes 3 Check if the codes are extremal 4 ⋆ Use subgroups of G Bernardo Rodrigues, rodrigues@ukzn.ac.za (University of KwaZulu-Natal) self-dual codes and rank 3 groups 06 August 2017 8 / 30
2-Transitive Automorphism Groups Table: Simple Socle dim n n 1 Socle 2 mod Extremal M 24 (Mathieu) 24 Golay yes HS (Higman-Sims) 176 none A n , n ≥ 5 n none PSL ( d , q ) , d ≥ 2 4 possib. none PSU ( 3 , 7 ) 344 none PSL ( 2 , 7 3 ) 344 GQR code no PSp ( 2 d , 2 ) 6 possib. none n ≤ 104 2 PSL ( 2 , p ) p + 1 QR-codes A n n none 1 8 | n , n ≤ 3928 2 QR codes Bernardo Rodrigues, rodrigues@ukzn.ac.za (University of KwaZulu-Natal) self-dual codes and rank 3 groups 06 August 2017 9 / 30
2-Transitive Automorphism Groups Extremal self-dual codes with a 2-transitive group have been classified In A. Malevich and W. Willems, On the classification of the extremal self-dual codes over small fields with 2 -transitive automorphism groups Des. Codes Cryptogr. 70 (2014), 69â ˘ A ¸ S76 showed that Theorem 2 Extremal codes C with 2 -transitive automorphism are known: (i) QR codes of length 8 , 24 , 32 , 48 , 80 or 104 ; (ii) Reed-Muller code of length 32 ; (iii) Possibly a code of length n = 1024 with E ⋊ PSL ( 2 , 2 5 ) ≤ Aut ( C ) Bernardo Rodrigues, rodrigues@ukzn.ac.za (University of KwaZulu-Natal) self-dual codes and rank 3 groups 06 August 2017 10 / 30
Finally in N. Chigira, M. Harada and M. Kitazume. On the classification of extremal doubly even self-dual codes with 2-transitive automorphism groups Des. Codes Cryptogr. 73 (2014), 33â ˘ A ¸ S35. showed that in fact Theorem 3 There is no extremal self-dual code of length 1024 . Bernardo Rodrigues, rodrigues@ukzn.ac.za (University of KwaZulu-Natal) self-dual codes and rank 3 groups 06 August 2017 11 / 30
Results on automorphism groups of self-dual codes Chigira, Harada and Kitazume in N. Chigira, M. Harada and M. Kitazume, Permutation groups and binary self-orthogonal codes. J. Algebra, 309 (2007), 610-621 proposed a way of constructing self-orthogonal codes from permutation groups Result 4.1 (Chigira, Harada and Kitazume, 2007) If there exists a self-dual code C, then C ( G , Ω) ⊥ ⊂ C ⊂ C ( G , Ω) . In particular, the code � Fix ( β ) | β ∈ I ( G ) � is self-orthogonal. The code C ( G , Ω) invariant under a permutation group G on an n -element set Ω is defined as C ( G , Ω) = � Fix ( β ) | β ∈ I ( G ) � ⊥ , where I ( G ) corresponds to the set of involutions of G and Fix ( β ) is the set of fixed points of β on Ω . Bernardo Rodrigues, rodrigues@ukzn.ac.za (University of KwaZulu-Natal) self-dual codes and rank 3 groups 06 August 2017 12 / 30
Günther and Nebe, in A. Günther and G. Nebe., Automorphisms of doubly even self-dual codes. Bull. London Math. Soc. , 41 (2009), 769-778 showed that Result 4.2 (Günther and Nebe, 2009) Let G ≤ S n and k = F 2 . Then there exists a self-dual code C ≤ k n with G ≤ Aut ( C ) if and only if every self-dual simple kG-module U occurs in the kG-module k n with even multiplicity. The next result deals with the existence of self-dual doubly-even codes invariant under permutation groups. Bernardo Rodrigues, rodrigues@ukzn.ac.za (University of KwaZulu-Natal) self-dual codes and rank 3 groups 06 August 2017 13 / 30
Result 4.3 (Günther and Nebe, 2009) Let G ≤ S n and k = F 2 . Then there is a self-dual doubly even code C = C ⊥ ≤ k n with G ≤ Aut ( C ) if and only if the following three conditions are fulfilled: (i) 8 | n; (ii) every self-dual composition factor of the kG-module k n occurs with even multiplicity; (iii) G ≤ A n . Bernardo Rodrigues, rodrigues@ukzn.ac.za (University of KwaZulu-Natal) self-dual codes and rank 3 groups 06 August 2017 14 / 30
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