A nice proof of Weis Duality Theorem Thomas Britz UNSW Sydney 1 0 - PowerPoint PPT Presentation

linear code rank function 1 2 3 4 5 1 0 1 0 0 0 1 1 0 0 ρ ( 124 ) = 3 0 0 0 1 1 Tutte polynomial � x ρ ( E ) − ρ ( A ) y | A |− ρ ( A ) T ( x + 1 , y + 1) = A ⊆ E = 6 + 9 x + 5 x 2 + x 3 + 5 y + y 2 + 4 xy + x 2 y

linear code rank function 1 2 3 4 5 1 0 1 0 0 0 1 1 0 0 ρ ( 124 ) = 3 0 0 0 1 1 Tutte polynomial and rank generator function � x ρ ( E ) − ρ ( A ) y | A |− ρ ( A ) T ( x + 1 , y + 1) = R ( x, y ) = A ⊆ E = 6 + 9 x + 5 x 2 + x 3 + 5 y + y 2 + 4 xy + x 2 y

linear code rank function 1 2 3 4 5 1 0 1 0 0 0 1 1 0 0 ρ ( 124 ) = 3 0 0 0 1 1 Tutte polynomial and rank generator function � x ρ ( E ) − ρ ( A ) y | A |− ρ ( A ) T ( x + 1 , y + 1) = R ( x, y ) = A ⊆ E = 6 + 9 x + 5 x 2 + x 3 + 5 y + y 2 + 4 xy + x 2 y ... and coboundary polynomial χ ( xy, y + 1 , 1) = y ρ ( E ) R ( x, y )

Tutte polynomial and rank generator function � x ρ ( E ) − ρ ( A ) y | A |− ρ ( A ) T ( x + 1 , y + 1) = R ( x, y ) = A ⊆ E ... and coboundary polynomial χ ( xy, y + 1 , 1) = y ρ ( E ) R ( x, y )

Tutte polynomial and rank generator function � x ρ ( E ) − ρ ( A ) y | A |− ρ ( A ) T ( x + 1 , y + 1) = R ( x, y ) = A ⊆ E ... and coboundary polynomial χ ( xy, y + 1 , 1) = y ρ ( E ) R ( x, y ) Lemma R C ⊥ ( x, y ) = R C ( y, x ) T C ⊥ ( x, y ) = T C ( y, x ) χ C ⊥ ( λ, x, y ) = λ − ρ C ( E ) χ C ( λ, x + ( λ − 1) y, x − y )

Tutte polynomial and rank generator function � x ρ ( E ) − ρ ( A ) y | A |− ρ ( A ) T ( x + 1 , y + 1) = R ( x, y ) = A ⊆ E ... and coboundary polynomial χ ( xy, y + 1 , 1) = y ρ ( E ) R ( x, y ) Lemma R C ⊥ ( x, y ) = R C ( y, x ) T C ⊥ ( x, y ) = T C ( y, x ) χ C ⊥ ( λ, x, y ) = λ − ρ C ( E ) χ C ( λ, x + ( λ − 1) y, x − y ) Proof ρ C ( E ) − ρ C ( A ) = | E − A | − ρ C ⊥ ( E − A ) �

MacWilliams Identity (1963) W C ⊥ ( x, y ) = 1 q k W C ( x + ( q − 1) y, x − y ) 1 0 1 0 0 W C ( x, y ) = x 5 + 4 x 3 y 2 + 3 xy 4 0 1 1 0 0 C 0 0 0 1 1 W C ⊥ ( x, y ) = 1 1 1 1 0 0 C ⊥ q k W C ( x + ( q − 1) y, x − y ) 0 0 0 1 1 = 1 1 + 4( x + y ) 3 ( x − y ) 2 + 4( x + y )( x − y ) 4 � � 2 3 = x 5 + x 3 y 2 + x 2 y 3 + y 5

MacWilliams Identity (1963) W C ⊥ ( x, y ) = 1 q k W C ( x + ( q − 1) y, x − y ) 1 0 1 0 0 W C ( x, y ) = x 5 + 4 x 3 y 2 + 3 xy 4 0 1 1 0 0 C 0 0 0 1 1 Greene’s Theorem (1976) W C ( x, y ) = χ C ( q, x, y )

MacWilliams Identity (1963) W C ⊥ ( x, y ) = 1 q k W C ( x + ( q − 1) y, x − y ) 1 0 1 0 0 W C ( x, y ) = χ (2 , x, y ) = x 5 + 4 x 3 y 2 + 3 xy 4 0 1 1 0 0 C 0 0 0 1 1 Greene’s Theorem (1976) W C ( x, y ) = χ C ( q, x, y )

MacWilliams Identity (1963) W C ⊥ ( x, y ) = 1 q k W C ( x + ( q − 1) y, x − y ) Proof Greene’s Theorem (1976) W C ( x, y ) = χ C ( q, x, y )

MacWilliams Identity (1963) W C ⊥ ( x, y ) = 1 q k W C ( x + ( q − 1) y, x − y ) Proof W C ⊥ ( x, y ) = χ C ⊥ ( q, x, y ) Greene’s Theorem (1976) W C ( x, y ) = χ C ( q, x, y )

MacWilliams Identity (1963) W C ⊥ ( x, y ) = 1 q k W C ( x + ( q − 1) y, x − y ) Proof W C ⊥ ( x, y ) = χ C ⊥ ( q, x, y ) = q − ρ ( E ) χ C ( q, x + ( q − 1) y, x − y ) Greene’s Theorem (1976) W C ( x, y ) = χ C ( q, x, y )

MacWilliams Identity (1963) W C ⊥ ( x, y ) = 1 q k W C ( x + ( q − 1) y, x − y ) Proof W C ⊥ ( x, y ) = χ C ⊥ ( q, x, y ) = q − ρ ( E ) χ C ( q, x + ( q − 1) y, x − y ) = 1 q k W C ( x + ( q − 1) y, x − y ) � Greene’s Theorem (1976) W C ( x, y ) = χ C ( q, x, y )

= a linear code over a finite field F q C Crapo Rota 1970 ρ C determines the codeword supports of C

= a linear code over a finite field F q C Crapo Rota 1970 ρ C determines the codeword supports of C “The Critical Theorem”

= a linear code over a finite field F q C Crapo Rota 1970 ρ C determines the codeword supports of C An infinite class of such results, eg. subcode supports Britz 2005

= a linear code over a finite field F q C Crapo Rota 1970 ρ C determines the codeword supports of C An infinite class of such results, eg. subcode supports Britz 2005 A small part of ρ C determines C ’s codeword weights Greene 1976

= a linear code over a finite field F q C Crapo Rota 1970 ρ C determines the codeword supports of C An infinite class of such results, eg. subcode supports Britz 2005 A small part of ρ C determines C ’s codeword weights Greene 1976 “Greene’s Theorem”

= a linear code over a finite field F q C Crapo Rota 1970 ρ C determines the codeword supports of C An infinite class of such results, eg. subcode supports Britz 2005 A small part of ρ C determines C ’s codeword weights Greene 1976 The same small part determines C ’s subcode weights Britz 2007

= a linear code over a finite field F q C Crapo Rota 1970 ρ C determines the codeword supports of C An infinite class of such results, eg. subcode supports Britz 2005 A small part of ρ C determines C ’s codeword weights Greene 1976 The same small part determines C ’s subcode weights Britz 2007 Tutte polynomial and subcode weights are equivalent Britz 2010

= a linear code over a finite field F q C Crapo Rota 1970 ρ C determines the codeword supports of C An infinite class of such results, eg. subcode supports Britz 2005 A small part of ρ C determines C ’s codeword weights Greene 1976 The same small part determines C ’s subcode weights Britz 2007 Tutte polynomial and subcode weights are equivalent Britz 2010 Skorabogatov 1992 ρ C does not determine the covering radius of C

= a linear code over a finite field F q C Crapo Rota 1970 ρ C determines the codeword supports of C An infinite class of such results, eg. subcode supports Britz 2005 A small part of ρ C determines C ’s codeword weights Greene 1976 The same small part determines C ’s subcode weights Britz 2007 Tutte polynomial and subcode weights are equivalent Britz 2010 Skorabogatov 1992 ρ C does not determine the covering radius of C Other properties of C not determined by ρ C BR 2005

= a linear code over a finite field F q C Crapo Rota 1970 ρ C determines the codeword supports of C An infinite class of such results, eg. subcode supports Britz 2005 A small part of ρ C determines C ’s codeword weights Greene 1976 The same small part determines C ’s subcode weights Britz 2007 Tutte polynomial and subcode weights are equivalent Britz 2010 Skorabogatov 1992 ρ C does not determine the covering radius of C Other properties of C not determined by ρ C BR 2005 The codeword weights of C determine those of C ⊥ MacWilliams 1963

= a linear code over a finite field F q C Crapo Rota 1970 ρ C determines the codeword supports of C An infinite class of such results, eg. subcode supports Britz 2005 A small part of ρ C determines C ’s codeword weights Greene 1976 The same small part determines C ’s subcode weights Britz 2007 Tutte polynomial and subcode weights are equivalent Britz 2010 Skorabogatov 1992 ρ C does not determine the covering radius of C Other properties of C not determined by ρ C BR 2005 The codeword weights of C determine those of C ⊥ MacWilliams 1963 “The MacWilliams’ Identity”

= a linear code over a finite field F q C Crapo Rota 1970 ρ C determines the codeword supports of C An infinite class of such results, eg. subcode supports Britz 2005 A small part of ρ C determines C ’s codeword weights Greene 1976 The same small part determines C ’s subcode weights Britz 2007 Tutte polynomial and subcode weights are equivalent Britz 2010 Skorabogatov 1992 ρ C does not determine the covering radius of C Other properties of C not determined by ρ C BR 2005 The codeword weights of C determine those of C ⊥ MacWilliams 1963 An infinite class of MacWilliams-type results Britz 2005

= a linear code over a finite field F q C Crapo Rota 1970 ρ C determines the codeword supports of C An infinite class of such results, eg. subcode supports Britz 2005 A small part of ρ C determines C ’s codeword weights Greene 1976 The same small part determines C ’s subcode weights Britz 2007 Tutte polynomial and subcode weights are equivalent Britz 2010 Skorabogatov 1992 ρ C does not determine the covering radius of C Other properties of C not determined by ρ C BR 2005 The codeword weights of C determine those of C ⊥ MacWilliams 1963 An infinite class of MacWilliams-type results Britz 2005 A general MacWilliams-type result for matroids BS 2008

= a linear code over a finite field F q C Crapo Rota 1970 ρ C determines the codeword supports of C An infinite class of such results, eg. subcode supports Britz 2005 A small part of ρ C determines C ’s codeword weights Greene 1976 The same small part determines C ’s subcode weights Britz 2007 Tutte polynomial and subcode weights are equivalent Britz 2010 Skorabogatov 1992 ρ C does not determine the covering radius of C Other properties of C not determined by ρ C BR 2005 The codeword weights of C determine those of C ⊥ MacWilliams 1963 An infinite class of MacWilliams-type results Britz 2005 A general MacWilliams-type result for matroids BS 2008 Matroid extensions of [Delsarte 1972, Duursma 2003] BS 2008

= a linear code over a finite field F q C Crapo Rota 1970 ρ C determines the codeword supports of C An infinite class of such results, eg. subcode supports Britz 2005 A small part of ρ C determines C ’s codeword weights Greene 1976 The same small part determines C ’s subcode weights Britz 2007 Tutte polynomial and subcode weights are equivalent Britz 2010 Skorabogatov 1992 ρ C does not determine the covering radius of C Other properties of C not determined by ρ C BR 2005 The codeword weights of C determine those of C ⊥ MacWilliams 1963 An infinite class of MacWilliams-type results Britz 2005 A general MacWilliams-type result for matroids BS 2008 Matroid extensions of [Delsarte 1972, Duursma 2003] BS 2008 Assmus Mattson 1969 t -designs from codeword supports

= a linear code over a finite field F q C Crapo Rota 1970 ρ C determines the codeword supports of C An infinite class of such results, eg. subcode supports Britz 2005 A small part of ρ C determines C ’s codeword weights Greene 1976 The same small part determines C ’s subcode weights Britz 2007 Tutte polynomial and subcode weights are equivalent Britz 2010 Skorabogatov 1992 ρ C does not determine the covering radius of C Other properties of C not determined by ρ C BR 2005 The codeword weights of C determine those of C ⊥ MacWilliams 1963 An infinite class of MacWilliams-type results Britz 2005 A general MacWilliams-type result for matroids BS 2008 Matroid extensions of [Delsarte 1972, Duursma 2003] BS 2008 Assmus Mattson 1969 t -designs from codeword supports t -designs from matroids and subcode supports BS 2008, BRS 2009

= a linear code over a finite field F q C Crapo Rota 1970 ρ C determines the codeword supports of C An infinite class of such results, eg. subcode supports Britz 2005 A small part of ρ C determines C ’s codeword weights Greene 1976 The same small part determines C ’s subcode weights Britz 2007 Tutte polynomial and subcode weights are equivalent Britz 2010 Skorabogatov 1992 ρ C does not determine the covering radius of C Other properties of C not determined by ρ C BR 2005 The codeword weights of C determine those of C ⊥ MacWilliams 1963 An infinite class of MacWilliams-type results Britz 2005 A general MacWilliams-type result for matroids BS 2008 Matroid extensions of [Delsarte 1972, Duursma 2003] BS 2008 Assmus Mattson 1969 t -designs from codeword supports t -designs from matroids and subcode supports BS 2008, BRS 2009 Minimal subcode weights of C determine those of C ⊥ Wei 1991

= a linear code over a finite field F q C Crapo Rota 1970 ρ C determines the codeword supports of C An infinite class of such results, eg. subcode supports Britz 2005 A small part of ρ C determines C ’s codeword weights Greene 1976 The same small part determines C ’s subcode weights Britz 2007 Tutte polynomial and subcode weights are equivalent Britz 2010 Skorabogatov 1992 ρ C does not determine the covering radius of C Other properties of C not determined by ρ C BR 2005 The codeword weights of C determine those of C ⊥ MacWilliams 1963 An infinite class of MacWilliams-type results Britz 2005 A general MacWilliams-type result for matroids BS 2008 Matroid extensions of [Delsarte 1972, Duursma 2003] BS 2008 Assmus Mattson 1969 t -designs from codeword supports t -designs from matroids and subcode supports BS 2008, BRS 2009 Minimal subcode weights of C determine those of C ⊥ Wei 1991 “Wei’s Duality Theorem”

= a linear code over a finite field F q C Crapo Rota 1970 ρ C determines the codeword supports of C An infinite class of such results, eg. subcode supports Britz 2005 A small part of ρ C determines C ’s codeword weights Greene 1976 The same small part determines C ’s subcode weights Britz 2007 Tutte polynomial and subcode weights are equivalent Britz 2010 Skorabogatov 1992 ρ C does not determine the covering radius of C Other properties of C not determined by ρ C BR 2005 The codeword weights of C determine those of C ⊥ MacWilliams 1963 An infinite class of MacWilliams-type results Britz 2005 A general MacWilliams-type result for matroids BS 2008 Matroid extensions of [Delsarte 1972, Duursma 2003] BS 2008 Assmus Mattson 1969 t -designs from codeword supports t -designs from matroids and subcode supports BS 2008, BRS 2009 Minimal subcode weights of C determine those of C ⊥ Wei 1991 BJMS 2012 Matroid extensions of this result

= a linear code over a finite field F q C Crapo Rota 1970 ρ C determines the codeword supports of C An infinite class of such results, eg. subcode supports Britz 2005 A small part of ρ C determines C ’s codeword weights Greene 1976 The same small part determines C ’s subcode weights Britz 2007 Tutte polynomial and subcode weights are equivalent Britz 2010 Skorabogatov 1992 ρ C does not determine the covering radius of C Other properties of C not determined by ρ C BR 2005 The codeword weights of C determine those of C ⊥ MacWilliams 1963 An infinite class of MacWilliams-type results Britz 2005 A general MacWilliams-type result for matroids BS 2008 Matroid extensions of [Delsarte 1972, Duursma 2003] BS 2008 Assmus Mattson 1969 t -designs from codeword supports t -designs from matroids and subcode supports BS 2008, BRS 2009 Minimal subcode weights of C determine those of C ⊥ Wei 1991 BJMS 2012 Matroid extensions of this result DESD [48,24,12] code higher weight enumerators BBSS 2007

= a linear code over a finite field F q C Crapo Rota 1970 ρ C determines the codeword supports of C An infinite class of such results, eg. subcode supports Britz 2005 A small part of ρ C determines C ’s codeword weights Greene 1976 The same small part determines C ’s subcode weights Britz 2007 Tutte polynomial and subcode weights are equivalent Britz 2010 Skorabogatov 1992 ρ C does not determine the covering radius of C Other properties of C not determined by ρ C BR 2005 The codeword weights of C determine those of C ⊥ MacWilliams 1963 An infinite class of MacWilliams-type results Britz 2005 A general MacWilliams-type result for matroids BS 2008 Matroid extensions of [Delsarte 1972, Duursma 2003] BS 2008 Assmus Mattson 1969 t -designs from codeword supports t -designs from matroids and subcode supports BS 2008, BRS 2009 Minimal subcode weights of C determine those of C ⊥ Wei 1991 BJMS 2012 Matroid extensions of this result DESD [48,24,12] code higher weight enumerators BBSS 2007

= a linear code over a finite field F q C Crapo Rota 1970 ρ C determines the codeword supports of C An infinite class of such results, eg. subcode supports Britz 2005 A small part of ρ C determines C ’s codeword weights Greene 1976 The same small part determines C ’s subcode weights Britz 2007 Tutte polynomial and subcode weights are equivalent Britz 2010 Skorabogatov 1992 ρ C does not determine the covering radius of C Other properties of C not determined by ρ C BR 2005 The codeword weights of C determine those of C ⊥ MacWilliams 1963 An infinite class of MacWilliams-type results Britz 2005 A general MacWilliams-type result for matroids BS 2008 Matroid extensions of [Delsarte 1972, Duursma 2003] BS 2008 Assmus Mattson 1969 t -designs from codeword supports t -designs from matroids and subcode supports BS 2008, BRS 2009 Minimal subcode weights of C determine those of C ⊥ Wei 1991 BJMS 2012 Matroid extensions of this result DESD [48,24,12] code higher weight enumerators BBSS 2007 CGB 2010, CGB 2013, BJM 2014, BSW 2015, BC Related results

linear code 1 0 1 0 0 0 1 1 0 0 0 0 0 1 1

linear code 1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 1 1 0 0 0 1 0 1 1 1 0 1 1 1 1 1 1 0 1 1 subcodes

linear code 1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 1 1 0 0 0 1 0 1 1 1 0 1 1 1 1 1 1 0 1 1 2 2 2 2 4 4 4 subcodes

linear code 1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 1 1 0 0 0 1 0 1 1 1 0 1 1 1 1 1 1 0 1 1 2 2 2 2 4 4 4 subcodes 1 0 1 0 0 1 0 1 0 0 1 0 1 0 0 1 1 0 0 0 1 1 0 0 0 0 0 0 1 1 0 1 1 0 0 0 1 1 0 0 0 0 0 1 1 0 1 1 1 1 0 0 0 1 1 1 0 1 1 1 0 1 1 0 0 1 0 1 1 1

linear code 1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 1 1 0 0 0 1 0 1 1 1 0 1 1 1 1 1 1 0 1 1 2 2 2 2 4 4 4 subcodes 1 0 1 0 0 1 0 1 0 0 1 0 1 0 0 1 1 0 0 0 1 1 0 0 0 0 0 0 1 1 0 1 1 0 0 0 1 1 0 0 0 0 0 1 1 0 1 1 1 1 0 0 0 1 1 1 0 1 1 1 0 1 1 0 0 1 0 1 1 1 3 4 5 4 5 4 5

linear code 1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 1 1 0 0 0 1 0 1 1 1 0 1 1 1 1 1 1 0 1 1 2 2 2 2 4 4 4 subcodes 1 0 1 0 0 1 0 1 0 0 1 0 1 0 0 1 1 0 0 0 1 1 0 0 0 0 0 0 1 1 0 1 1 0 0 0 1 1 0 0 0 0 0 1 1 0 1 1 1 1 0 0 0 1 1 1 0 1 1 1 0 1 1 0 0 1 0 1 1 1 3 4 5 4 5 4 5 1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 5

linear code higher weights d 1 = 1 0 1 0 0 d 2 = 0 1 1 0 0 0 0 0 1 1 d 3 = 1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 1 1 0 0 0 1 0 1 1 1 0 1 1 1 1 1 1 0 1 1 2 2 2 2 4 4 4 subcodes 1 0 1 0 0 1 0 1 0 0 1 0 1 0 0 1 1 0 0 0 1 1 0 0 0 0 0 0 1 1 0 1 1 0 0 0 1 1 0 0 0 0 0 1 1 0 1 1 1 1 0 0 0 1 1 1 0 1 1 1 0 1 1 0 0 1 0 1 1 1 3 4 5 4 5 4 5 1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 5

linear code higher weights d 1 = 2 1 0 1 0 0 d 2 = 0 1 1 0 0 0 0 0 1 1 d 3 = 1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 1 1 0 0 0 1 0 1 1 1 0 1 1 1 1 1 1 0 1 1 2 2 2 2 4 4 4 subcodes 1 0 1 0 0 1 0 1 0 0 1 0 1 0 0 1 1 0 0 0 1 1 0 0 0 0 0 0 1 1 0 1 1 0 0 0 1 1 0 0 0 0 0 1 1 0 1 1 1 1 0 0 0 1 1 1 0 1 1 1 0 1 1 0 0 1 0 1 1 1 3 4 5 4 5 4 5 1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 5

linear code higher weights d 1 = 2 1 0 1 0 0 d 2 = 3 0 1 1 0 0 0 0 0 1 1 d 3 = 1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 1 1 0 0 0 1 0 1 1 1 0 1 1 1 1 1 1 0 1 1 2 2 2 2 4 4 4 subcodes 1 0 1 0 0 1 0 1 0 0 1 0 1 0 0 1 1 0 0 0 1 1 0 0 0 0 0 0 1 1 0 1 1 0 0 0 1 1 0 0 0 0 0 1 1 0 1 1 1 1 0 0 0 1 1 1 0 1 1 1 0 1 1 0 0 1 0 1 1 1 3 4 5 4 5 4 5 1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 5

linear code higher weights d 1 = 2 1 0 1 0 0 d 2 = 3 0 1 1 0 0 0 0 0 1 1 d 3 = 5 1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 1 1 0 0 0 1 0 1 1 1 0 1 1 1 1 1 1 0 1 1 2 2 2 2 4 4 4 subcodes 1 0 1 0 0 1 0 1 0 0 1 0 1 0 0 1 1 0 0 0 1 1 0 0 0 0 0 0 1 1 0 1 1 0 0 0 1 1 0 0 0 0 0 1 1 0 1 1 1 1 0 0 0 1 1 1 0 1 1 1 0 1 1 0 0 1 0 1 1 1 3 4 5 4 5 4 5 1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 5

linear code higher weights d ⊥ d 1 = 2 1 = 2 1 0 1 0 0 d ⊥ d 2 = 3 2 = 5 0 1 1 0 0 0 0 0 1 1 d 3 = 5 1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 1 1 0 0 0 1 0 1 1 1 0 1 1 1 1 1 1 0 1 1 2 2 2 2 4 4 4 subcodes 1 0 1 0 0 1 0 1 0 0 1 0 1 0 0 1 1 0 0 0 1 1 0 0 0 0 0 0 1 1 0 1 1 0 0 0 1 1 0 0 0 0 0 1 1 0 1 1 1 1 0 0 0 1 1 1 0 1 1 1 0 1 1 0 0 1 0 1 1 1 3 4 5 4 5 4 5 1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 5