Automorphisms of extremal codes Gabriele Nebe Lehrstuhl D f ur - PowerPoint PPT Presentation

Automorphisms of extremal codes Gabriele Nebe Lehrstuhl D f¨ ur Mathematik ALCOMA 15

Plan The use of symmetry ◮ Beautiful objects have symmetries. ◮ Symmetries help to reduce the search space for nice objects ◮ and hence make huge problems acessible to computations. The use of challenge problems ◮ Applications for classical theories and theorems such as ◮ Burnside orbit counting ◮ Invariant theory of finite groups ◮ Theory of quadratic forms ◮ Representation theory of finite groups ◮ Provide a practical introduction to abstract theory.

Self-dual codes Definition ◮ A linear binary code C of length n is a subspace C ≤ F n 2 . ◮ The dual code of C is C ⊥ := { x ∈ F n 2 | ( x, c ) := � n i =1 x i c i = 0 for all c ∈ C } ◮ C is called self-dual if C = C ⊥ . ◮ Aut( C ) = { σ ∈ S n | σ ( C ) = C } . Facts ◮ dim( C ) + dim( C ⊥ ) = n so C = C ⊥ ⇒ dim( C ) = n 2 . ◮ Let 1 = (1 , . . . , 1) . Then ( c, c ) = ( c, 1 ) . ◮ So if C = C ⊥ then 1 ∈ C .

Doubly-even self-dual codes The Hamming weight. ◮ The Hamming weight of a codeword c ∈ C is wt( c ) := |{ i | c i � = 0 }| . ◮ wt( c ) ≡ 2 ( c, c ) , so C ⊆ C ⊥ implies wt( C ) ⊂ 2 Z . ◮ C is called doubly-even if wt( C ) ⊂ 4 Z . ◮ Fact: C = C ⊥ ≤ F n 2 doubly-even ⇒ n ∈ 8 Z . ◮ The minimum distance d ( C ) := min { wt( c ) | 0 � = c ∈ C } . ◮ A self-dual code C ≤ F n 2 is called extremal if d ( C ) = 4 + 4 ⌊ n 24 ⌋ . ◮ The weight enumerator of C is p C := � c ∈ C x n − wt( c ) y wt( c ) ∈ C [ x, y ] n .

Examples for self-dual doubly-even codes Hamming Code   1 0 0 0 0 1 1 1 0 1 0 0 1 0 1 1   h 8 :   0 0 1 0 1 1 0 1   0 0 0 1 1 1 1 0 the extended Hamming code, the unique doubly-even self-dual code of length 8, p h 8 ( x, y ) = x 8 + 14 x 4 y 4 + y 8 and Aut( h 8 ) = 2 3 : L 3 (2) . Golay Code The binary Golay code G 24 is the unique doubly-even self-dual code of length 24 with minimum distance ≥ 8 . Aut( G 24 ) = M 24 p G 24 = x 24 + 759 x 16 y 8 + 2576 x 12 y 12 + 759 x 8 y 16 + y 24

Application of invariant theory c ∈ C x n − wt( c ) y wt( c ) ∈ C [ x, y ] n . The weight enumerator of C is p C := � Theorem (Gleason, ICM 1970) Let C = C ⊥ ≤ F n 2 be doubly even. Then d ( C ) ≤ 4 + 4 ⌊ n 24 ⌋ Doubly-even self-dual codes achieving equality are called extremal.

Application of invariant theory c ∈ C x n − wt( c ) y wt( c ) ∈ C [ x, y ] n . The weight enumerator of C is p C := � Theorem (Gleason, ICM 1970) Let C = C ⊥ ≤ F n 2 be doubly even. Then d ( C ) ≤ 4 + 4 ⌊ n 24 ⌋ Doubly-even self-dual codes achieving equality are called extremal. Proof: ◮ p C ( x, y ) = p C ( x, iy ) , p C ( x, y ) = p C ⊥ ( x, y ) = p C ( x + y 2 , x − y 2 ) √ √ � 1 � 1 � � 0 1 1 ◮ G 192 := � , � . √ 0 i 1 − 1 2 ◮ p C ∈ Inv( G 192 ) = C [ p h 8 , p G 24 ] ◮ ∃ ! f ∈ C [ p h 8 , p G 24 ] 8 m such that f (1 , y ) = 1 + 0 y 4 + . . . + 0 y 4 ⌊ m 3 ⌋ + a m y 4 ⌊ m 3 ⌋ +4 + b m y 4 ⌊ m 3 ⌋ +8 + . . . ◮ a m > 0 for all m

Application of invariant theory c ∈ C x n − wt( c ) y wt( c ) ∈ C [ x, y ] n . The weight enumerator of C is p C := � Theorem (Gleason, ICM 1970) Let C = C ⊥ ≤ F n 2 be doubly even. Then d ( C ) ≤ 4 + 4 ⌊ n 24 ⌋ Doubly-even self-dual codes achieving equality are called extremal. Proof: ◮ p C ( x, y ) = p C ( x, iy ) , p C ( x, y ) = p C ⊥ ( x, y ) = p C ( x + y 2 , x − y 2 ) √ √ � 1 � 1 � � 0 1 1 ◮ G 192 := � , � . √ 0 i 1 − 1 2 ◮ p C ∈ Inv( G 192 ) = C [ p h 8 , p G 24 ] ◮ ∃ ! f ∈ C [ p h 8 , p G 24 ] 8 m such that f (1 , y ) = 1 + 0 y 4 + . . . + 0 y 4 ⌊ m 3 ⌋ + a m y 4 ⌊ m 3 ⌋ +4 + b m y 4 ⌊ m 3 ⌋ +8 + . . . ◮ a m > 0 for all m Proposition b m < 0 for all m ≥ 494 so there is no extremal code of length ≥ 3952 .

Automorphism groups of extremal codes length 8 24 32 40 48 72 80 96 104 ≥ 3952 d ( C ) 4 8 8 8 12 16 16 20 20 extremal h 8 G 24 5 16 , 470 QR 48 ? ≥ 15 ? ≥ 1 0 Aut( C ) = { σ ∈ S n | σ ( C ) = C } is the automorphism group of C ≤ F n 2 . ◮ 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? ◮ If C is such a (72 , 36 , 16) code then Aut( C ) has order ≤ 5 .

Automorphism groups of extremal codes length 8 24 32 40 48 72 80 96 104 ≥ 3952 d ( C ) 4 8 8 8 12 16 16 20 20 extremal h 8 G 24 5 16 , 470 QR 48 ? ≥ 15 ? ≥ 1 0 Aut( C ) = { σ ∈ S n | σ ( C ) = C } is the automorphism group of C ≤ F n 2 . ◮ 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? ◮ If C is such a (72 , 36 , 16) code then Aut( C ) has order ≤ 5 . ◮ There is no beautiful (72 , 36 , 16) self-dual code.

The Type of an automorphism Definition Let σ ∈ S n of prime order p . Then σ is of Type ( z, f ) , if σ has z p -cycles and f fixed points. zp + f = n . ◮ Let p be odd, σ = (1 , 2 , .., p )( p + 1 , .., 2 p ) ... (( z − 1) p + 1 , .., zp ) . 2 = Fix( σ ) ⊥ E ( σ ) ∼ = F z + f ⊥ F z ( p − 1) ◮ F n with 2 2 1 . . . 1 0 . . . 0 . . . 0 . . . 0 0 0 . . . 0 0 . . . 0 1 . . . 1 . . . 0 . . . 0 0 0 . . . 0 0 . . . 0 0 . . . 0 . . . 1 . . . 1 0 0 . . . 0 Fix( σ ) = � 0 . . . 0 0 . . . 0 . . . 0 . . . 0 1 0 . . . 0 � 0 . . . 0 0 . . . 0 . . . 0 . . . 0 0 1 . . . 0 0 . . . 0 0 . . . 0 . . . 0 . . . 0 0 0 . . . 1 � �� � � �� � � �� � p p p E ( σ ) = Fix( σ ) ⊥ = { ( x 1 , . . . , x p , x p +1 , . . . , x 2 p , . . . , x ( z − 1) p +1 , . . . , x zp , 0 , . . . , 0) | x 1 + . . . + x p = x p +1 + . . . + x 2 p = . . . = x ( z − 1) p +1 + . . . + x zp = 0 }

Two self-dual codes of smaller length ◮ Let C ≤ F n 2 and p an odd prime, ◮ σ = (1 , 2 , .., p )( p + 1 , .., 2 p ) ... (( z − 1) p + 1 , .., zp ) ∈ Aut( C ) . ◮ Then C = C ∩ Fix( σ ) ⊕ C ∩ E ( σ ) =: Fix C ( σ ) ⊕ E C ( σ ) . c zp +1 . . . c n ) ∈ C } ∼ Fix C ( σ ) = { ( c p . . . c p c 2 p . . . c 2 p . . . c zp . . . c zp = � �� � � �� � � �� � p p p = { ( c p c 2 p . . . c zp c zp +1 . . . c n ) ∈ F z + f π (Fix C ( σ )) | c ∈ Fix C ( σ ) } 2 ◮ and C ⊥ = C ⊥ ∩ Fix( σ ) ⊕ C ⊥ ∩ E ( σ ) . ◮ C = C ⊥ then Fix C ( σ ) is self-dual in Fix( σ ) and E C ( σ ) is (Hermitian) self-dual in E ( σ ) . Fact π (Fix C ( σ )) is a self-dual code of length z + f , in particular dim(Fix C ( σ )) = z + f and | Fix C ( σ ) | = 2 ( z + f ) / 2 . 2

Application of Burnside’s orbit counting theorem Theorem (Conway, Pless, 1982) Let C = C ⊥ ≤ F n 2 , σ ∈ Aut( C ) of odd prime order p and Type ( z, f ) . 2 ( z + f ) / 2 ≡ 2 n/ 2 Then (mod p ) . Proof: Apply orbit counting: � 1 The number of G -orbits on a finite set M is g ∈ G | Fix M ( g ) | . | G | Here G = � σ � , M = C , Fix C ( g ) = Fix C ( σ ) for all 1 � = g ∈ G , and the p (2 n/ 2 + ( p − 1)2 ( z + f ) / 2 ) ∈ N . number of � σ � -orbits on C is 1 Corollary C = C ⊥ ≤ F n 2 , p > n/ 2 an odd prime divisor of | Aut( C ) | , then p ≡ ± 1 (mod 8) . Here z = 1 , f = n − p , ( z + f ) / 2 = ( n − ( p − 1)) / 2 , so 2 ( p − 1) / 2 is 1 mod p and hence 2 must be a square modulo p .

Application of quadratic forms Remark ◮ C = C ⊥ ⇒ 1 = (1 , . . . , 1) ∈ C , since ( c, c ) = ( c, 1 ) . ◮ If C is self-dual then n = 2 dim( C ) is even and 1 ∈ C ⊥ = C ⊂ 1 ⊥ = { c ∈ F n 2 | wt( c ) even } . ◮ Self-dual doubly-even codes correspond to totally isotropic subspaces in the quadratic space E n − 2 := ( 1 ⊥ / � 1 � , q ) , q ( c + � 1 � ) = 1 2 wt( c ) (mod 2) ∈ F 2 . ◮ C = C ⊥ ≤ F n 2 doubly-even ⇒ n ∈ 8 Z . Theorem (A. Meyer, N. 2009) Let C = C ⊥ ≤ F n 2 doubly-even. Then Aut( C ) ≤ Alt n .

Application of quadratic forms: Some background ◮ Assume n ∈ 8 Z . ◮ E n − 2 := ( 1 ⊥ / � 1 � , q ) , q ( c + � 1 � ) = 1 2 wt( c ) (mod 2) ∈ F 2 . is an ( n − 2) -dimensional quadratic space over F 2 . ◮ There is X ≤ E n − 2 with X = X ⊥ and q ( X ) = { 0 } call such X self-dual isotropic. ◮ C = C ⊥ ≤ F n 2 , doubly-even, then X = C/ � 1 � ≤ E n − 2 is self-dual isotropic. ◮ O ( E n − 2 ) = { g ∈ GL( E n − 2 ) | q ( g ( x )) = q ( x ) for all x ∈ E n − 2 } the orthogonal group of E n − 2 . Definition Fix X 0 ≤ E n − 2 self-dual isotropic. D : O ( E n − 2 ) → { 1 , − 1 } , D ( g ) := ( − 1) dim( X 0 / ( X 0 ∩ g ( X 0 ))) the Dickson invariant. Fact g ∈ Stab O ( E n − 2 ) ( X ) ⇒ D ( g ) = 1 .