Symmetric rank distance codes Kai-Uwe Schmidt Otto-von-Guericke University Magdeburg, Germany 1
Rank distance codes Rank distance code: A set of (possibly restricted) matrices over F q with the property that every nonzero difference has rank at least some fixed number d . 2
Rank distance codes Rank distance code: A set of (possibly restricted) matrices over F q with the property that every nonzero difference has rank at least some fixed number d . Typical questions: How large can such a set be? 2
Rank distance codes Rank distance code: A set of (possibly restricted) matrices over F q with the property that every nonzero difference has rank at least some fixed number d . Typical questions: How large can such a set be? How can we construct them? 2
Rank distance codes Rank distance code: A set of (possibly restricted) matrices over F q with the property that every nonzero difference has rank at least some fixed number d . Typical questions: How large can such a set be? How can we construct them? What can we say about the (rank) distance distribution? 2
Rank distance codes Rank distance code: A set of (possibly restricted) matrices over F q with the property that every nonzero difference has rank at least some fixed number d . Typical questions: How large can such a set be? How can we construct them? What can we say about the (rank) distance distribution? Why? Array codes, network codes, classical coding theory, semifields, planar functions, APN functions, and more. 2
Reed-Muller codes RM( r , m ): Polynomials in F 2 [ x 1 , . . . , x m ] / ( x 2 1 − x 1 , . . . , x 2 m − x m ) of degree at most r . 3
Reed-Muller codes RM( r , m ): Polynomials in F 2 [ x 1 , . . . , x m ] / ( x 2 1 − x 1 , . . . , x 2 m − x m ) of degree at most r . The cosets RM(2 , m ) / RM(1 , m ) are given by quadratic forms � a ij x i x j i < j or alternating matrices. 3
Reed-Muller codes RM( r , m ): Polynomials in F 2 [ x 1 , . . . , x m ] / ( x 2 1 − x 1 , . . . , x 2 m − x m ) of degree at most r . The cosets RM(2 , m ) / RM(1 , m ) are given by quadratic forms � a ij x i x j i < j or alternating matrices. The weight distribution of such a coset depends only on the rank r of this matrix (always even) and the minimum weight is 2 m − 1 − 2 m − r / 2 − 1 . 3
Kerdock codes What is the maximum number of m × m alternating matrices over F 2 such that every nonzero difference has rank m (even)? 4
Kerdock codes What is the maximum number of m × m alternating matrices over F 2 such that every nonzero difference has rank m (even)? Answer: 2 m − 1 (the matrices must have distinct first rows). 4
Kerdock codes What is the maximum number of m × m alternating matrices over F 2 such that every nonzero difference has rank m (even)? Answer: 2 m − 1 (the matrices must have distinct first rows). This gives the Kerdock code. (Kerdock, 1972) 4
Kerdock codes What is the maximum number of m × m alternating matrices over F 2 such that every nonzero difference has rank m (even)? Answer: 2 m − 1 (the matrices must have distinct first rows). This gives the Kerdock code. (Kerdock, 1972) . . . 4
Kerdock codes What is the maximum number of m × m alternating matrices over F 2 such that every nonzero difference has rank m (even)? Answer: 2 m − 1 (the matrices must have distinct first rows). This gives the Kerdock code. (Kerdock, 1972) . . . 4
Beyond Kerdock codes What is the maximum number of m × m alternating matrices over F 2 such that every nonzero difference has rank at least 2 d ? 5
Beyond Kerdock codes What is the maximum number of m × m alternating matrices over F 2 such that every nonzero difference has rank at least 2 d ? Answer: Known, but not so easy. Need association schemes. (Cameron & Seidel, 1973), (Delsarte & Goethals, 1975). 5
Beyond Kerdock codes What is the maximum number of m × m alternating matrices over F 2 such that every nonzero difference has rank at least 2 d ? Answer: Known, but not so easy. Need association schemes. (Cameron & Seidel, 1973), (Delsarte & Goethals, 1975). There is a Singleton-type bound and in case of equality: The set is an equivalent of an MDS code. There is a Reed-Solomon-like construction. The distance distribution is uniquely determined. This gives the Delsarte-Goethals codes. 5
How does it generalise? GRM( r , m ): Polynomials in F q [ x 1 , . . . , x m ] / ( x q 1 − x 1 , . . . , x q m − x m ) of degree at most r . 6
How does it generalise? GRM( r , m ): Polynomials in F q [ x 1 , . . . , x m ] / ( x q 1 − x 1 , . . . , x q m − x m ) of degree at most r . For q > 2, the cosets GRM(2 , m ) / GRM(1 , m ) are given by quadratic forms � a ij x i x j , i , j or equivalently by the cosets of alternating matrices. 6
How does it generalise? GRM( r , m ): Polynomials in F q [ x 1 , . . . , x m ] / ( x q 1 − x 1 , . . . , x q m − x m ) of degree at most r . For q > 2, the cosets GRM(2 , m ) / GRM(1 , m ) are given by quadratic forms � a ij x i x j , i , j or equivalently by the cosets of alternating matrices. Orbits under the general linear group: [ A ] ∼ [ B ] if and only if [ A ] = [ L T BL ] for an invertible L . 6
Another viewpoint Orbits of symmetric matrices under the general linear group: A ∼ B if and only if A = L T BL for an invertible L . 7
Another viewpoint Orbits of symmetric matrices under the general linear group: A ∼ B if and only if A = L T BL for an invertible L . The associated quadratic forms � � b ii x 2 x T B x = 2 b ij x i x j + i i < j i 7
Another viewpoint Orbits of symmetric matrices under the general linear group: A ∼ B if and only if A = L T BL for an invertible L . The associated quadratic forms � � b ii x 2 x T B x = 2 b ij x i x j + i i < j i are the ordinary quadratic forms for odd q ; 7
Another viewpoint Orbits of symmetric matrices under the general linear group: A ∼ B if and only if A = L T BL for an invertible L . The associated quadratic forms � � b ii x 2 x T B x = 2 b ij x i x j + i i < j i are the ordinary quadratic forms for odd q ; the quadratic forms over a Galois ring for even q . 7
Connections to coding theory Forms on F m q Codes q Alternating 2 RM(2 , m ) / RM(1 , m ) 8
Connections to coding theory Forms on F m q Codes q Alternating 2 RM(2 , m ) / RM(1 , m ) Quadratic all GRM(2 , m ) / GRM(1 , m ) 8
Connections to coding theory Forms on F m q Codes q Alternating 2 RM(2 , m ) / RM(1 , m ) Quadratic all GRM(2 , m ) / GRM(1 , m ) Symmetric odd GRM(2 , m ) / GRM(1 , m ) 8
Connections to coding theory Forms on F m q Codes q Alternating 2 RM(2 , m ) / RM(1 , m ) Quadratic all GRM(2 , m ) / GRM(1 , m ) Symmetric odd GRM(2 , m ) / GRM(1 , m ) Symmetric even ZRM(2 , m ) / ZRM(1 , m ) 8
Association schemes Association scheme: a set of points X and a partition { R 0 , R 1 , . . . , R n } of X × X satisfying certain conditions. 9
Association schemes Association scheme: a set of points X and a partition { R 0 , R 1 , . . . , R n } of X × X satisfying certain conditions. A translation scheme is an association scheme, where ( X , +) is an abelian group and there is a partition ( X i ) of X such that R i = { ( A , B ) : A − B ∈ X i } . 9
Association schemes Association scheme: a set of points X and a partition { R 0 , R 1 , . . . , R n } of X × X satisfying certain conditions. A translation scheme is an association scheme, where ( X , +) is an abelian group and there is a partition ( X i ) of X such that R i = { ( A , B ) : A − B ∈ X i } . There is a partition � X 0 , � X 1 , . . . , � X n of the character group � X of X such that � is constant for all ψ ∈ � ψ ( A ) X k . A ∈ X i 9
Association schemes Association scheme: a set of points X and a partition { R 0 , R 1 , . . . , R n } of X × X satisfying certain conditions. A translation scheme is an association scheme, where ( X , +) is an abelian group and there is a partition ( X i ) of X such that R i = { ( A , B ) : A − B ∈ X i } . There is a partition � X 0 , � X 1 , . . . , � X n of the character group � X of X such that � is constant for all ψ ∈ � ψ ( A ) X k . A ∈ X i This partition defines an association scheme on � X and is called the dual translation scheme. 9
Families of translation schemes Classical schemes: Hamming, bilinear forms, alternating forms, Hermitian forms. 10
Families of translation schemes Classical schemes: Hamming, bilinear forms, alternating forms, Hermitian forms. The orbits of the quadratic forms and the symmetric bilinear forms give rise to translation schemes, called Qua( m , q ) and Sym( m , q ), respectively. 10
Families of translation schemes Classical schemes: Hamming, bilinear forms, alternating forms, Hermitian forms. The orbits of the quadratic forms and the symmetric bilinear forms give rise to translation schemes, called Qua( m , q ) and Sym( m , q ), respectively. They are not polynomial and generally not symmetric. 10
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