the coset leader weight enumerator of the code of the
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The coset leader weight enumerator of the code of the twisted cubic - PowerPoint PPT Presentation

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The coset leader weight enumerator of the code of the twisted cubic Ruud Pellikaan g.r.pellikaan@tue.nl Arithmetic, Geometry, Cryptography and Coding Theory AGC 2 T 17, Luminy, 11 June 2019 Faculteit Wiskunde & Informatica Gilles

  1. Projective systems, codes and minimal weight 16/53 PROPOSITION Let C be a nondegenerate [ n , k ] code over F q with generator matrix G Let c be a nonzero codeword c = m G for the unique m ∈ F k q Let H be the hyperplane in P k − 1 ( F q ) with equation H : m 1 X 1 + · · · + m k X k = 0 Then n − wt ( c ) is equal to the number of points of of P G in H And ¯ A w is the number of hyperplanes in the projective space P k − 1 ( F q ) with exactly n − w points of P P on it Faculteit Wiskunde & Informatica

  2. Projective systems, codes and minimal weight 16/53 PROPOSITION Let C be a nondegenerate [ n , k ] code over F q with generator matrix G Let c be a nonzero codeword c = m G for the unique m ∈ F k q Let H be the hyperplane in P k − 1 ( F q ) with equation H : m 1 X 1 + · · · + m k X k = 0 Then n − wt ( c ) is equal to the number of points of of P G in H And ¯ A w is the number of hyperplanes in the projective space P k − 1 ( F q ) with exactly n − w points of P P on it Faculteit Wiskunde & Informatica

  3. Projective systems, codes and minimal weight 16/53 PROPOSITION Let C be a nondegenerate [ n , k ] code over F q with generator matrix G Let c be a nonzero codeword c = m G for the unique m ∈ F k q Let H be the hyperplane in P k − 1 ( F q ) with equation H : m 1 X 1 + · · · + m k X k = 0 Then n − wt ( c ) is equal to the number of points of of P G in H And ¯ A w is the number of hyperplanes in the projective space P k − 1 ( F q ) with exactly n − w points of P P on it Faculteit Wiskunde & Informatica

  4. Points in general position - MDS codes 17/53 Let C be a nondegenerate [ n , k ] q code Then C is an MDS code, that is an [ n , k , n − k + 1 ] q code attaining the Singleton bound if and only if the points of the projective system P G in P k − 1 ( F q ) are in general position that is to say that there are at most k − 1 points of P G in a hyperplane Faculteit Wiskunde & Informatica

  5. Points in general position - MDS codes 17/53 Let C be a nondegenerate [ n , k ] q code Then C is an MDS code, that is an [ n , k , n − k + 1 ] q code attaining the Singleton bound if and only if the points of the projective system P G in P k − 1 ( F q ) are in general position that is to say that there are at most k − 1 points of P G in a hyperplane Faculteit Wiskunde & Informatica

  6. Points in general position - MDS codes 17/53 Let C be a nondegenerate [ n , k ] q code Then C is an MDS code, that is an [ n , k , n − k + 1 ] q code attaining the Singleton bound if and only if the points of the projective system P G in P k − 1 ( F q ) are in general position that is to say that there are at most k − 1 points of P G in a hyperplane Faculteit Wiskunde & Informatica

  7. Dually: arrangements and codes 18/53 An arrangement ( H 1 , . . . , H n ) is an n -tuple of hyperplanes in F k q or P r ( F q ) such that their intersection is { 0 } or empty, resp. Let G = ( g ij ) be a generator matrix of a nondegenerate [ n , k ] code C So G has no zero columns q or P k − 1 ( F q ) with equation Let H j be the linear hyperplane in F k g 1 j X 1 + · · · + g kj X k = 0 . Let A G be the arrangement ( H 1 , . . . , H n ) associated with G Faculteit Wiskunde & Informatica

  8. Dually: arrangements and codes 18/53 An arrangement ( H 1 , . . . , H n ) is an n -tuple of hyperplanes in F k q or P r ( F q ) such that their intersection is { 0 } or empty, resp. Let G = ( g ij ) be a generator matrix of a nondegenerate [ n , k ] code C So G has no zero columns q or P k − 1 ( F q ) with equation Let H j be the linear hyperplane in F k g 1 j X 1 + · · · + g kj X k = 0 . Let A G be the arrangement ( H 1 , . . . , H n ) associated with G Faculteit Wiskunde & Informatica

  9. Dually: arrangements and codes 18/53 An arrangement ( H 1 , . . . , H n ) is an n -tuple of hyperplanes in F k q or P r ( F q ) such that their intersection is { 0 } or empty, resp. Let G = ( g ij ) be a generator matrix of a nondegenerate [ n , k ] code C So G has no zero columns q or P k − 1 ( F q ) with equation Let H j be the linear hyperplane in F k g 1 j X 1 + · · · + g kj X k = 0 . Let A G be the arrangement ( H 1 , . . . , H n ) associated with G Faculteit Wiskunde & Informatica

  10. Arrangements, codes and minimal weight 19/53 PROPOSITION Let C be a nondegenerate [ n , k ] code over F q with generator matrix G Let c be a codeword c = x G for the unique x ∈ F k q Then n − wt ( c ) is equal to the number of hyperplanes of A G going through ( x 1 : · · · : x k ) And ¯ A w is the number of points in P k − 1 ( F q ) on exactly n − w hyperplanes of A G Faculteit Wiskunde & Informatica

  11. Arrangements, codes and minimal weight 19/53 PROPOSITION Let C be a nondegenerate [ n , k ] code over F q with generator matrix G Let c be a codeword c = x G for the unique x ∈ F k q Then n − wt ( c ) is equal to the number of hyperplanes of A G going through ( x 1 : · · · : x k ) And ¯ A w is the number of points in P k − 1 ( F q ) on exactly n − w hyperplanes of A G Faculteit Wiskunde & Informatica

  12. Arrangements, codes and minimal weight 19/53 PROPOSITION Let C be a nondegenerate [ n , k ] code over F q with generator matrix G Let c be a codeword c = x G for the unique x ∈ F k q Then n − wt ( c ) is equal to the number of hyperplanes of A G going through ( x 1 : · · · : x k ) And ¯ A w is the number of points in P k − 1 ( F q ) on exactly n − w hyperplanes of A G Faculteit Wiskunde & Informatica

  13. Projective system versus arrangement of lines 20/53 Figuur: Projective system (L), Arrangement of lines (R) in P 2 ( F q ) of [ 4 , 3 , 2 ] code Faculteit Wiskunde & Informatica

  14. ¯ A n and complement of hyperplanes 21/53 In particular ¯ A n is equal to the number of points that is in the complement of the union of these hyperplanes in P k − 1 ( F q ) This number can be computed by the principle of inclusion/exclusion A n = q k − 1 ¯ q − 1 − | H 1 ∪ · · · ∪ H n | = n � ( − 1 ) w � | H i 1 ∩ · · · ∩ H i w | w = 0 i 1 < ··· < i w Faculteit Wiskunde & Informatica

  15. ¯ A n and complement of hyperplanes 21/53 In particular ¯ A n is equal to the number of points that is in the complement of the union of these hyperplanes in P k − 1 ( F q ) This number can be computed by the principle of inclusion/exclusion A n = q k − 1 ¯ q − 1 − | H 1 ∪ · · · ∪ H n | = n � ( − 1 ) w � | H i 1 ∩ · · · ∩ H i w | w = 0 i 1 < ··· < i w Faculteit Wiskunde & Informatica

  16. REMARK 22/53 Define for a subset J of { 1 , 2 , . . . , n } C ( J ) = { c ∈ C | c j = 0 for all j ∈ J } The encoding map x �→ x G = c from vectors x ∈ F k q to codewords gives the following isomorphism of vector spaces H j ∼ � = C ( J ) j ∈ J Faculteit Wiskunde & Informatica

  17. REMARK 22/53 Define for a subset J of { 1 , 2 , . . . , n } C ( J ) = { c ∈ C | c j = 0 for all j ∈ J } The encoding map x �→ x G = c from vectors x ∈ F k q to codewords gives the following isomorphism of vector spaces H j ∼ � = C ( J ) j ∈ J Faculteit Wiskunde & Informatica

  18. DEFINITION 23/53 Define following Katsman and Tsfasman l ( J ) = dim C ( J ) B J = q l ( J ) − 1 � B t = B J | J | = t Then B J is equal to the number of nonzero codewords c that are zero at all j in J and This is equal to the number of nonzero elements of the intersection � H j j ∈ J Faculteit Wiskunde & Informatica

  19. DEFINITION 23/53 Define following Katsman and Tsfasman l ( J ) = dim C ( J ) B J = q l ( J ) − 1 � B t = B J | J | = t Then B J is equal to the number of nonzero codewords c that are zero at all j in J and This is equal to the number of nonzero elements of the intersection � H j j ∈ J Faculteit Wiskunde & Informatica

  20. DEFINITION 23/53 Define following Katsman and Tsfasman l ( J ) = dim C ( J ) B J = q l ( J ) − 1 � B t = B J | J | = t Then B J is equal to the number of nonzero codewords c that are zero at all j in J and This is equal to the number of nonzero elements of the intersection � H j j ∈ J Faculteit Wiskunde & Informatica

  21. DEFINITION extended version 24/53 B J ( T ) = T l ( J ) − 1 � B t ( T ) = B J ( T ) | J | = t Faculteit Wiskunde & Informatica

  22. PROPOSITION 25/53 The following relation between the B t and A w holds n − t � n − w � � B t = A w t w = d and for the extended version n − t � n − w � � B t ( T ) = A w ( T ) t w = d Faculteit Wiskunde & Informatica

  23. PROPOSITION 25/53 The following relation between the B t and A w holds n − t � n − w � � B t = A w t w = d and for the extended version n − t � n − w � � B t ( T ) = A w ( T ) t w = d Faculteit Wiskunde & Informatica

  24. THEOREM (Katsman-Tsfasman and Jurrius-P) 26/53 The homogeneous weight enumerator of C can be expressed in terms of the B t as follows n W C ( X , Y ) = X n + � B t ( X − Y ) t Y n − t t = 0 and for the extended version n W C ( X , Y , T ) = X n + � B t ( T )( X − Y ) t Y n − t t = 0 This motivic version works over any field of coefficients The number of codewords in C ⊗ F q m of weight w is A w ( q m ) and W C ( X , Y , q m ) = W C ⊗ F qm ( X , Y ) Faculteit Wiskunde & Informatica

  25. THEOREM (Katsman-Tsfasman and Jurrius-P) 26/53 The homogeneous weight enumerator of C can be expressed in terms of the B t as follows n W C ( X , Y ) = X n + � B t ( X − Y ) t Y n − t t = 0 and for the extended version n W C ( X , Y , T ) = X n + � B t ( T )( X − Y ) t Y n − t t = 0 This motivic version works over any field of coefficients The number of codewords in C ⊗ F q m of weight w is A w ( q m ) and W C ( X , Y , q m ) = W C ⊗ F qm ( X , Y ) Faculteit Wiskunde & Informatica

  26. THEOREM (Katsman-Tsfasman and Jurrius-P) 26/53 The homogeneous weight enumerator of C can be expressed in terms of the B t as follows n W C ( X , Y ) = X n + � B t ( X − Y ) t Y n − t t = 0 and for the extended version n W C ( X , Y , T ) = X n + � B t ( T )( X − Y ) t Y n − t t = 0 This motivic version works over any field of coefficients The number of codewords in C ⊗ F q m of weight w is A w ( q m ) and W C ( X , Y , q m ) = W C ⊗ F qm ( X , Y ) Faculteit Wiskunde & Informatica

  27. PROPOSITION 27/53 The weight distribution of an MDS code of length n and dimension k is given for w ≥ d = n − k + 1 by � w − d � n � w � � q w − d + 1 − j − 1 � � ( − 1 ) j A w = w j j = 0 and for the extend version � w − d � n � w � � T w − d + 1 − j − 1 � � ( − 1 ) j A w ( T ) = w j j = 0 Faculteit Wiskunde & Informatica

  28. PROPOSITION 27/53 The weight distribution of an MDS code of length n and dimension k is given for w ≥ d = n − k + 1 by � w − d � n � w � � q w − d + 1 − j − 1 � � ( − 1 ) j A w = w j j = 0 and for the extend version � w − d � n � w � � T w − d + 1 − j − 1 � � ( − 1 ) j A w ( T ) = w j j = 0 Faculteit Wiskunde & Informatica

  29. Arrangement of 4 lines of [ 4 , 3 , 2 ] code 28/53 Faculteit Wiskunde & Informatica

  30. Connections 29/53 The following polynomials determine each other: W C ( X , Y , T ) extended weight enumerator of C { W ( r ) C ( X , Y ) : r = 1 , . . . , k } generalized weight enumerators of C t C ( X , Y ) dichromatic Tutte polynomial of matroid M C by Greene χ C ( S , T ) coboundary or two variable char.pol. of geometric lattice L C ζ C ( S , T ) two variable zeta function of C by Duursma But W C ( X , Y ) is weaker than W C ( X , Y , T ) Faculteit Wiskunde & Informatica

  31. Connections 29/53 The following polynomials determine each other: W C ( X , Y , T ) extended weight enumerator of C { W ( r ) C ( X , Y ) : r = 1 , . . . , k } generalized weight enumerators of C t C ( X , Y ) dichromatic Tutte polynomial of matroid M C by Greene χ C ( S , T ) coboundary or two variable char.pol. of geometric lattice L C ζ C ( S , T ) two variable zeta function of C by Duursma But W C ( X , Y ) is weaker than W C ( X , Y , T ) Faculteit Wiskunde & Informatica

  32. Connections 29/53 The following polynomials determine each other: W C ( X , Y , T ) extended weight enumerator of C { W ( r ) C ( X , Y ) : r = 1 , . . . , k } generalized weight enumerators of C t C ( X , Y ) dichromatic Tutte polynomial of matroid M C by Greene χ C ( S , T ) coboundary or two variable char.pol. of geometric lattice L C ζ C ( S , T ) two variable zeta function of C by Duursma But W C ( X , Y ) is weaker than W C ( X , Y , T ) Faculteit Wiskunde & Informatica

  33. Connections 29/53 The following polynomials determine each other: W C ( X , Y , T ) extended weight enumerator of C { W ( r ) C ( X , Y ) : r = 1 , . . . , k } generalized weight enumerators of C t C ( X , Y ) dichromatic Tutte polynomial of matroid M C by Greene χ C ( S , T ) coboundary or two variable char.pol. of geometric lattice L C ζ C ( S , T ) two variable zeta function of C by Duursma But W C ( X , Y ) is weaker than W C ( X , Y , T ) Faculteit Wiskunde & Informatica

  34. Connections 29/53 The following polynomials determine each other: W C ( X , Y , T ) extended weight enumerator of C { W ( r ) C ( X , Y ) : r = 1 , . . . , k } generalized weight enumerators of C t C ( X , Y ) dichromatic Tutte polynomial of matroid M C by Greene χ C ( S , T ) coboundary or two variable char.pol. of geometric lattice L C ζ C ( S , T ) two variable zeta function of C by Duursma But W C ( X , Y ) is weaker than W C ( X , Y , T ) Faculteit Wiskunde & Informatica

  35. Connections 29/53 The following polynomials determine each other: W C ( X , Y , T ) extended weight enumerator of C { W ( r ) C ( X , Y ) : r = 1 , . . . , k } generalized weight enumerators of C t C ( X , Y ) dichromatic Tutte polynomial of matroid M C by Greene χ C ( S , T ) coboundary or two variable char.pol. of geometric lattice L C ζ C ( S , T ) two variable zeta function of C by Duursma But W C ( X , Y ) is weaker than W C ( X , Y , T ) Faculteit Wiskunde & Informatica

  36. 30/53 Coset leader weight enumerator Helleseth, Jurrius-P, Utomo-P Faculteit Wiskunde & Informatica

  37. Coset leader weight enumerator 31/53 Let C be a linear [ n , k , d ] q code The weight of the coset y + C is defined by wt ( y + C ) = min { wt ( y + c ) : c ∈ C } A coset leader of r + C is a choice of an element of minimal weight in the coset r + C Let α i = the number of cosets of C that are of weight i The coset leader weight enumerator of C is the polynomial defined by n � α i X n − i Y i α C ( X , Y ) = i = 0 Faculteit Wiskunde & Informatica

  38. Coset leader weight enumerator 31/53 Let C be a linear [ n , k , d ] q code The weight of the coset y + C is defined by wt ( y + C ) = min { wt ( y + c ) : c ∈ C } A coset leader of r + C is a choice of an element of minimal weight in the coset r + C Let α i = the number of cosets of C that are of weight i The coset leader weight enumerator of C is the polynomial defined by n � α i X n − i Y i α C ( X , Y ) = i = 0 Faculteit Wiskunde & Informatica

  39. Coset leader weight enumerator 31/53 Let C be a linear [ n , k , d ] q code The weight of the coset y + C is defined by wt ( y + C ) = min { wt ( y + c ) : c ∈ C } A coset leader of r + C is a choice of an element of minimal weight in the coset r + C Let α i = the number of cosets of C that are of weight i The coset leader weight enumerator of C is the polynomial defined by n � α i X n − i Y i α C ( X , Y ) = i = 0 Faculteit Wiskunde & Informatica

  40. Coset leader weight enumerator 31/53 Let C be a linear [ n , k , d ] q code The weight of the coset y + C is defined by wt ( y + C ) = min { wt ( y + c ) : c ∈ C } A coset leader of r + C is a choice of an element of minimal weight in the coset r + C Let α i = the number of cosets of C that are of weight i The coset leader weight enumerator of C is the polynomial defined by n � α i X n − i Y i α C ( X , Y ) = i = 0 Faculteit Wiskunde & Informatica

  41. Coset leader decoding 32/53 The coset leader decoder D is defined by - Preprocessing: make a list of all coset leaders - Input: r a received word - Let e be the chosen coset leader of r + C in the list - Output: D ( r ) = c = r − e Then c ∈ C and d ( r , c ) = wt ( e ) = d ( r , C ) Hence D is a nearest codeword decoder Note that c is not necessarily the codeword sent Faculteit Wiskunde & Informatica

  42. Coset leader decoding 32/53 The coset leader decoder D is defined by - Preprocessing: make a list of all coset leaders - Input: r a received word - Let e be the chosen coset leader of r + C in the list - Output: D ( r ) = c = r − e Then c ∈ C and d ( r , c ) = wt ( e ) = d ( r , C ) Hence D is a nearest codeword decoder Note that c is not necessarily the codeword sent Faculteit Wiskunde & Informatica

  43. Coset leader decoding 32/53 The coset leader decoder D is defined by - Preprocessing: make a list of all coset leaders - Input: r a received word - Let e be the chosen coset leader of r + C in the list - Output: D ( r ) = c = r − e Then c ∈ C and d ( r , c ) = wt ( e ) = d ( r , C ) Hence D is a nearest codeword decoder Note that c is not necessarily the codeword sent Faculteit Wiskunde & Informatica

  44. P C , dc for coset leader decoder 33/53 PROPOSITION The probability of decoding correctly of the coset leader decoder on a q -ary symmetric channel with cross-over probability p is given by � � p P C , dc ( p ) = α C 1 − p , q − 1 Faculteit Wiskunde & Informatica

  45. Properties coset leader weight enumerator 34/53 Let C be a linear [ n , k , d ] q code with covering radius ρ ( C ) Then � n � ( q − 1 ) i if i ≤ ( d − 1 ) / 2 α i = i Since every vector e of weight at most ( d − 1 ) / 2 is the unique word of minimal weight in the coset e + C α i = 0 if i > ρ ( C ) Since by definition there is no word r such that d ( r , C ) > ρ ( C ) n � α i = q n − k α C ( 1 , 1 ) = i = 0 Since the total number of cosets is q n − k Faculteit Wiskunde & Informatica

  46. Properties coset leader weight enumerator 34/53 Let C be a linear [ n , k , d ] q code with covering radius ρ ( C ) Then � n � ( q − 1 ) i if i ≤ ( d − 1 ) / 2 α i = i Since every vector e of weight at most ( d − 1 ) / 2 is the unique word of minimal weight in the coset e + C α i = 0 if i > ρ ( C ) Since by definition there is no word r such that d ( r , C ) > ρ ( C ) n � α i = q n − k α C ( 1 , 1 ) = i = 0 Since the total number of cosets is q n − k Faculteit Wiskunde & Informatica

  47. Properties coset leader weight enumerator 34/53 Let C be a linear [ n , k , d ] q code with covering radius ρ ( C ) Then � n � ( q − 1 ) i if i ≤ ( d − 1 ) / 2 α i = i Since every vector e of weight at most ( d − 1 ) / 2 is the unique word of minimal weight in the coset e + C α i = 0 if i > ρ ( C ) Since by definition there is no word r such that d ( r , C ) > ρ ( C ) n � α i = q n − k α C ( 1 , 1 ) = i = 0 Since the total number of cosets is q n − k Faculteit Wiskunde & Informatica

  48. C n the dual repetition code 35/53 Let C n be the dual code of the n -fold repetition code So ( 1 , 1 , . . . , 1 ) is a parity check matrix of C n And C n is an [ n , n − 1 , 2 ] q code and we can choose the ( λ, 0 , . . . , 0 ) for λ ∈ F q as a complete collection of coset leaders Hence the coset leader weight enumerator of C n is given by α C n ( X , Y ) = X n + ( q − 1 ) X n − 1 Y 1 Faculteit Wiskunde & Informatica

  49. C n the dual repetition code 35/53 Let C n be the dual code of the n -fold repetition code So ( 1 , 1 , . . . , 1 ) is a parity check matrix of C n And C n is an [ n , n − 1 , 2 ] q code and we can choose the ( λ, 0 , . . . , 0 ) for λ ∈ F q as a complete collection of coset leaders Hence the coset leader weight enumerator of C n is given by α C n ( X , Y ) = X n + ( q − 1 ) X n − 1 Y 1 Faculteit Wiskunde & Informatica

  50. C n the dual repetition code 35/53 Let C n be the dual code of the n -fold repetition code So ( 1 , 1 , . . . , 1 ) is a parity check matrix of C n And C n is an [ n , n − 1 , 2 ] q code and we can choose the ( λ, 0 , . . . , 0 ) for λ ∈ F q as a complete collection of coset leaders Hence the coset leader weight enumerator of C n is given by α C n ( X , Y ) = X n + ( q − 1 ) X n − 1 Y 1 Faculteit Wiskunde & Informatica

  51. C n the dual repetition code 35/53 Let C n be the dual code of the n -fold repetition code So ( 1 , 1 , . . . , 1 ) is a parity check matrix of C n And C n is an [ n , n − 1 , 2 ] q code and we can choose the ( λ, 0 , . . . , 0 ) for λ ∈ F q as a complete collection of coset leaders Hence the coset leader weight enumerator of C n is given by α C n ( X , Y ) = X n + ( q − 1 ) X n − 1 Y 1 Faculteit Wiskunde & Informatica

  52. C m ⊗ C n product code 36/53 Let C m ⊗ C n be the product code of C m and C n Its codewords are considered as m × n matrices with entries in F q such that every row sum is zero and every column sum is zero Then C m ⊗ C n is an [ mn , ( m − 1 )( n − 1 ) , 4 ] q code Its coset leader weight enumerator is determined for q = 2 and q = 3 by Utomo-P But it is an open question for other q Faculteit Wiskunde & Informatica

  53. C m ⊗ C n product code 36/53 Let C m ⊗ C n be the product code of C m and C n Its codewords are considered as m × n matrices with entries in F q such that every row sum is zero and every column sum is zero Then C m ⊗ C n is an [ mn , ( m − 1 )( n − 1 ) , 4 ] q code Its coset leader weight enumerator is determined for q = 2 and q = 3 by Utomo-P But it is an open question for other q Faculteit Wiskunde & Informatica

  54. C m ⊗ C n product code 36/53 Let C m ⊗ C n be the product code of C m and C n Its codewords are considered as m × n matrices with entries in F q such that every row sum is zero and every column sum is zero Then C m ⊗ C n is an [ mn , ( m − 1 )( n − 1 ) , 4 ] q code Its coset leader weight enumerator is determined for q = 2 and q = 3 by Utomo-P But it is an open question for other q Faculteit Wiskunde & Informatica

  55. Extended coset leader weight enumerator 37/53 PROPOSITION (Helleseth, Jurrius-P) Let C be a linear [ n , k , d ] q code Then there exist polynomials α i ( T ) such that α i ( q m ) = the number of cosets of C ⊗ F q m that are of weight i α i ( T ) is divisible by T − 1 for i > 0 Define ¯ α i ( T ) = α i ( T ) / ( T − 1 ) The extended coset leader weight enumerator of C is the polynomial defined by n � α i ( T ) X n − i Y i α C ( X , Y , T ) = i = 0 Faculteit Wiskunde & Informatica

  56. Extended coset leader weight enumerator 37/53 PROPOSITION (Helleseth, Jurrius-P) Let C be a linear [ n , k , d ] q code Then there exist polynomials α i ( T ) such that α i ( q m ) = the number of cosets of C ⊗ F q m that are of weight i α i ( T ) is divisible by T − 1 for i > 0 Define ¯ α i ( T ) = α i ( T ) / ( T − 1 ) The extended coset leader weight enumerator of C is the polynomial defined by n � α i ( T ) X n − i Y i α C ( X , Y , T ) = i = 0 Faculteit Wiskunde & Informatica

  57. Extended coset leader weight enumerator 37/53 PROPOSITION (Helleseth, Jurrius-P) Let C be a linear [ n , k , d ] q code Then there exist polynomials α i ( T ) such that α i ( q m ) = the number of cosets of C ⊗ F q m that are of weight i α i ( T ) is divisible by T − 1 for i > 0 Define ¯ α i ( T ) = α i ( T ) / ( T − 1 ) The extended coset leader weight enumerator of C is the polynomial defined by n � α i ( T ) X n − i Y i α C ( X , Y , T ) = i = 0 Faculteit Wiskunde & Informatica

  58. Coset versus syndrome 38/53 Let C be a linear [ n , k , d ] q code let H be a parity check matrix of C and r ∈ F n q Then r 1 + C = r 2 + C if and only if H r T 1 = H r T 2 Then the column vector s = H r T ∈ F n − k q is called the syndrome of r with respect to H Hence there is a one-one correspondence between cosets of C and syndromes in F n − k q Faculteit Wiskunde & Informatica

  59. Coset versus syndrome 38/53 Let C be a linear [ n , k , d ] q code let H be a parity check matrix of C and r ∈ F n q Then r 1 + C = r 2 + C if and only if H r T 1 = H r T 2 Then the column vector s = H r T ∈ F n − k q is called the syndrome of r with respect to H Hence there is a one-one correspondence between cosets of C and syndromes in F n − k q Faculteit Wiskunde & Informatica

  60. Syndrome weight 39/53 Let H be a parity check matrix of a linear [ n , k ] code C over F q The weight of s with respect to H also called the syndrome weight of s is defined by wt H ( s ) = wt ( r + C ) A syndrome s is a linear combination of the columns of H The syndrome weight of of s is the minimal way to write s as a linear combination of the columns of a parity check matrix Hence α i is the number of vectors that are in the span of i columns of H but not in the span of i − 1 columns of H Faculteit Wiskunde & Informatica

  61. Syndrome weight 39/53 Let H be a parity check matrix of a linear [ n , k ] code C over F q The weight of s with respect to H also called the syndrome weight of s is defined by wt H ( s ) = wt ( r + C ) A syndrome s is a linear combination of the columns of H The syndrome weight of of s is the minimal way to write s as a linear combination of the columns of a parity check matrix Hence α i is the number of vectors that are in the span of i columns of H but not in the span of i − 1 columns of H Faculteit Wiskunde & Informatica

  62. Syndrome weight 39/53 Let H be a parity check matrix of a linear [ n , k ] code C over F q The weight of s with respect to H also called the syndrome weight of s is defined by wt H ( s ) = wt ( r + C ) A syndrome s is a linear combination of the columns of H The syndrome weight of of s is the minimal way to write s as a linear combination of the columns of a parity check matrix Hence α i is the number of vectors that are in the span of i columns of H but not in the span of i − 1 columns of H Faculteit Wiskunde & Informatica

  63. Syndrome weight 39/53 Let H be a parity check matrix of a linear [ n , k ] code C over F q The weight of s with respect to H also called the syndrome weight of s is defined by wt H ( s ) = wt ( r + C ) A syndrome s is a linear combination of the columns of H The syndrome weight of of s is the minimal way to write s as a linear combination of the columns of a parity check matrix Hence α i is the number of vectors that are in the span of i columns of H but not in the span of i − 1 columns of H Faculteit Wiskunde & Informatica

  64. Syndrome weight 39/53 Let H be a parity check matrix of a linear [ n , k ] code C over F q The weight of s with respect to H also called the syndrome weight of s is defined by wt H ( s ) = wt ( r + C ) A syndrome s is a linear combination of the columns of H The syndrome weight of of s is the minimal way to write s as a linear combination of the columns of a parity check matrix Hence α i is the number of vectors that are in the span of i columns of H but not in the span of i − 1 columns of H Faculteit Wiskunde & Informatica

  65. Projective systems of H of [ 6 , 3 , 3 ] codes 40/53 Figuur: Two projective systems that induce the same geometric lattice, but induce codes with different coset leader weight enumerators Derived arrangement of projective system Faculteit Wiskunde & Informatica

  66. Derived arrangement of H of [ 4 , 1 , 4 ] code 41/53 Faculteit Wiskunde & Informatica

  67. 42/53 Normal Rational Curve Segre, .... , Bruen-Hirschfeld, Blokhuis-P-Sz˝ onyi Faculteit Wiskunde & Informatica

  68. Normal rational curve C r 43/53 The normal rational curve of degree r is the curve C r in P r with parametric representation ( s r : s r − 1 t : . . . : st r − 1 : t r ) with ( s : t ) ∈ P 1 Alternatively given by the vanishing ideal I ( C r ) that is generated by the 2 × 2 minors of the 2 × r matrix � X 0 � X 1 . . . X i . . . X r − 1 . X 1 X 2 . . . X i + 1 . . . X r C 2 is the irreducible conic in P 2 C 3 is the twisted conic in P 3 Faculteit Wiskunde & Informatica

  69. Normal rational curve C r 43/53 The normal rational curve of degree r is the curve C r in P r with parametric representation ( s r : s r − 1 t : . . . : st r − 1 : t r ) with ( s : t ) ∈ P 1 Alternatively given by the vanishing ideal I ( C r ) that is generated by the 2 × 2 minors of the 2 × r matrix � X 0 � X 1 . . . X i . . . X r − 1 . X 1 X 2 . . . X i + 1 . . . X r C 2 is the irreducible conic in P 2 C 3 is the twisted conic in P 3 Faculteit Wiskunde & Informatica

  70. Normal rational curve C r 43/53 The normal rational curve of degree r is the curve C r in P r with parametric representation ( s r : s r − 1 t : . . . : st r − 1 : t r ) with ( s : t ) ∈ P 1 Alternatively given by the vanishing ideal I ( C r ) that is generated by the 2 × 2 minors of the 2 × r matrix � X 0 � X 1 . . . X i . . . X r − 1 . X 1 X 2 . . . X i + 1 . . . X r C 2 is the irreducible conic in P 2 C 3 is the twisted conic in P 3 Faculteit Wiskunde & Informatica

  71. Normal rational curve C r 43/53 The normal rational curve of degree r is the curve C r in P r with parametric representation ( s r : s r − 1 t : . . . : st r − 1 : t r ) with ( s : t ) ∈ P 1 Alternatively given by the vanishing ideal I ( C r ) that is generated by the 2 × 2 minors of the 2 × r matrix � X 0 � X 1 . . . X i . . . X r − 1 . X 1 X 2 . . . X i + 1 . . . X r C 2 is the irreducible conic in P 2 C 3 is the twisted conic in P 3 Faculteit Wiskunde & Informatica

  72. NRC and generalized Reed-Solomon codes 44/53 C r ( F q ) has q + 1 points lying in general position in P r ( F q ) The projective system of these q + 1 points in P r ( F q ) comes from a generalized Reed-Solomon (GRS) code with parameters [ q + 1 , r + 1 , q + 1 − r ] The dual code is again a generalized Reed-Solomon code with parameters [ q + 1 , q − r , r + 2 ] Faculteit Wiskunde & Informatica

  73. NRC and generalized Reed-Solomon codes 44/53 C r ( F q ) has q + 1 points lying in general position in P r ( F q ) The projective system of these q + 1 points in P r ( F q ) comes from a generalized Reed-Solomon (GRS) code with parameters [ q + 1 , r + 1 , q + 1 − r ] The dual code is again a generalized Reed-Solomon code with parameters [ q + 1 , q − r , r + 2 ] Faculteit Wiskunde & Informatica

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