Linear codes and Betti numbers of Stanley-Reisner rings associated - PowerPoint PPT Presentation

Stanley-Reisner rings associated to matroids Let M be a matroid on the ground set E . Let K be any field, and S = K [ X ] = K [ X e , e ∈ E ] . 10

Stanley-Reisner rings associated to matroids Let M be a matroid on the ground set E . Let K be any field, and S = K [ X ] = K [ X e , e ∈ E ] . Definition The Stanley-Reisner ideal of M is � � X σ = � I M = X e | σ ∈ C . e ∈ σ and the Stanley-Reisner ring R M = S / I M . 10

✲ ♦r ♠✉❧t✐❣r❛❞❡❞ ❇❡tt✐ ♥✉♠❜❡rs✿ ✲❣r❛❞❡❞ ❇❡tt✐ ♥✉♠❜❡rs✿ ❣❧♦❜❛❧ ❇❡tt✐ ♥✉♠❜❡rs✿ Betti numbers S / I M has a minimal free resolution 11

✲ ♦r ♠✉❧t✐❣r❛❞❡❞ ❇❡tt✐ ♥✉♠❜❡rs✿ ✲❣r❛❞❡❞ ❇❡tt✐ ♥✉♠❜❡rs✿ ❣❧♦❜❛❧ ❇❡tt✐ ♥✉♠❜❡rs✿ Betti numbers S / I M has a minimal free resolution 0 ← S / I M ← F 0 ← F 1 ← · · · ← F s ← 0 where � S ( − α ) β i ,α . F i = α ∈ N E 11

Betti numbers S / I M has a minimal free resolution 0 ← S / I M ← F 0 ← F 1 ← · · · ← F s ← 0 where � S ( − α ) β i ,α . F i = α ∈ N E 3 types of Betti numbers: N E ✲ ♦r ♠✉❧t✐❣r❛❞❡❞ ❇❡tt✐ ♥✉♠❜❡rs✿ β i ,α � N ✲❣r❛❞❡❞ ❇❡tt✐ ♥✉♠❜❡rs✿ β i , d = β i ,α | α | = d � ❣❧♦❜❛❧ ❇❡tt✐ ♥✉♠❜❡rs✿ β i = β i , d d � 0 11

Betti numbers S / I M has a minimal free resolution 0 ← S / I M ← F 0 ← F 1 ← · · · ← F s ← 0 where � S ( − α ) β i ,α . F i = α ∈ N E 3 types of Betti numbers: N E ✲ ♦r ♠✉❧t✐❣r❛❞❡❞ ❇❡tt✐ ♥✉♠❜❡rs✿ β i ,α � N ✲❣r❛❞❡❞ ❇❡tt✐ ♥✉♠❜❡rs✿ β i , d = β i ,α | α | = d � ❣❧♦❜❛❧ ❇❡tt✐ ♥✉♠❜❡rs✿ β i = β i , d d � 0 F 0 = S β 1 ,σ = 1 ⇔ σ ∈ C . 11

Betti table The Betti table of a matroid is a matrix together with an integer n where the number in the i -th column and the j -th row represents β i , i + j + c − 2 . The suffix c on the table denotes the minimal absolute values of a twist occuring. 12

Betti table The Betti table of a matroid is a matrix together with an integer n where the number in the i -th column and the j -th row represents β i , i + j + c − 2 . The suffix c on the table denotes the minimal absolute values of a twist occuring. Example Let M be the matroid with circuits {{ 1 , 2 , 4 } , { 1 , 2 , 3 } , { 3 , 4 }} . 12

Betti table The Betti table of a matroid is a matrix together with an integer n where the number in the i -th column and the j -th row represents β i , i + j + c − 2 . The suffix c on the table denotes the minimal absolute values of a twist occuring. Example Let M be the matroid with circuits {{ 1 , 2 , 4 } , { 1 , 2 , 3 } , { 3 , 4 }} . 0 ← R M ← S ← S ( − 2 ) ⊕ S ( − 3 ) 2 ← S ( − 4 ) 2 ← 0 12

Betti table The Betti table of a matroid is a matrix together with an integer n where the number in the i -th column and the j -th row represents β i , i + j + c − 2 . The suffix c on the table denotes the minimal absolute values of a twist occuring. Example Let M be the matroid with circuits {{ 1 , 2 , 4 } , { 1 , 2 , 3 } , { 3 , 4 }} . 0 ← R M ← S ← S ( − 2 ) ⊕ S ( − 3 ) 2 ← S ( − 4 ) 2 ← 0 Betti table: � 1 � 0 . 2 2 2 12

✐♠ Hochster’s formula Let M be a matroid on the ground set E . We give E any total order. 13

✐♠ Hochster’s formula Let M be a matroid on the ground set E . We give E any total order. The chain complex of M over K is ∂ r − 1 ∂ 0 ∂ 1 K ∂ r � � 0 ← K ← ← · · · ← ← · · · ← 0 . K F ∈ M F ∈ M # F = 1 # F = r 13

✐♠ Hochster’s formula Let M be a matroid on the ground set E . We give E any total order. The chain complex of M over K is ∂ r − 1 ∂ 0 ∂ 1 K ∂ r � � 0 ← K ← ← · · · ← ← · · · ← 0 . K F ∈ M F ∈ M # F = 1 # F = r The boundary maps are: if F = { x 0 < · · · < x i } , i � ( − 1 ) j e { x 0 , ··· , ˇ ∂ i ( e F ) = x j , ··· , x i } . j = 0 13

Hochster’s formula Let M be a matroid on the ground set E . We give E any total order. The chain complex of M over K is ∂ r − 1 ∂ 0 ∂ 1 K ∂ r � � 0 ← K ← ← · · · ← ← · · · ← 0 . K F ∈ M F ∈ M # F = 1 # F = r The boundary maps are: if F = { x 0 < · · · < x i } , i � ( − 1 ) j e { x 0 , ··· , ˇ ∂ i ( e F ) = x j , ··· , x i } . j = 0 Definition The i -th reduced homology of M over K is the K vector space ˜ H i ( M , K ) = ker ( ∂ i ) / ✐♠ ( ∂ i + 1 ) and its dimension is denoted by ˜ h i ( M , K ) . 13

✐❢ ♦t❤❡r✇✐s❡ Hochster’s formula Theorem β i ,σ ( K ) = ˜ h | σ |− i − 1 ( M | σ , K ) 14

Hochster’s formula Theorem β i ,σ ( K ) = ˜ h | σ |− i − 1 ( M | σ , K ) Theorem Let M be a matroid on E of rank r . Then � ( − 1 ) r χ ( M ) ✐❢ i = r − 1 ˜ h i ( M , K ) = 0 ♦t❤❡r✇✐s❡ 14

Hochster’s formula Theorem β i ,σ ( K ) = ˜ h | σ |− i − 1 ( M | σ , K ) Theorem Let M be a matroid on E of rank r . Then � ( − 1 ) r χ ( M ) ✐❢ i = r − 1 ˜ h i ( M , K ) = 0 ♦t❤❡r✇✐s❡ Corollary The Betti numbers of a matroid are independent of the field K . 14

✐❢ ♦t❤❡r✇✐s❡ Goal Look at relations between matroids, their Betti numbers and their generalized Hamming weights. 15

✐❢ ♦t❤❡r✇✐s❡ Goal Look at relations between matroids, their Betti numbers and their generalized Hamming weights. Remark 1:The N E -graded case: 15

Goal Look at relations between matroids, their Betti numbers and their generalized Hamming weights. Remark 1:The N E -graded case: � 1 ✐❢ σ ∈ C β 1 ,σ = 0 ♦t❤❡r✇✐s❡ 15

Outline 1 Matroids - Hamming weights - Betti numbers 16

Outline 1 Matroids - Hamming weights - Betti numbers Relation between the nullity function and 2 non-redundancy of circuits 16

Outline 1 Matroids - Hamming weights - Betti numbers Relation between the nullity function and 2 non-redundancy of circuits 3 Betti numbers and generalized Hamming weights 16

Outline 1 Matroids - Hamming weights - Betti numbers Relation between the nullity function and 2 non-redundancy of circuits 3 Betti numbers and generalized Hamming weights Dualities 4 16

Outline 1 Matroids - Hamming weights - Betti numbers Relation between the nullity function and 2 non-redundancy of circuits 3 Betti numbers and generalized Hamming weights Dualities 4 Examples and preliminary summary 5 16

Outline 1 Matroids - Hamming weights - Betti numbers Relation between the nullity function and 2 non-redundancy of circuits 3 Betti numbers and generalized Hamming weights Dualities 4 Examples and preliminary summary 5 Constant weight codes 6 16

Outline 1 Matroids - Hamming weights - Betti numbers Relation between the nullity function and 2 non-redundancy of circuits 3 Betti numbers and generalized Hamming weights Dualities 4 Examples and preliminary summary 5 Constant weight codes 6 Weight enumerators of matroids 7 16

Outline 1 Matroids - Hamming weights - Betti numbers Relation between the nullity function and 2 non-redundancy of circuits 3 Betti numbers and generalized Hamming weights Dualities 4 Examples and preliminary summary 5 Constant weight codes 6 Weight enumerators of matroids 7 Algebraic geometric codes 8 16

Content 1 Matroids - Hamming weights - Betti numbers Relation between the nullity function and 2 non-redundancy of circuits 3 Betti numbers and generalized Hamming weights Dualities 4 Examples and preliminary summary 5 Constant weight codes 6 Weight enumerators of matroids 7 Algebraic geometric codes 8 17

♥♦♥✲r❡❞✉♥❞❛♥t ❛♥❞ Non-redundancy An ingredient in understanding the role of cirquits for the nullity of subsets of E . Definition Let M be a matroid, and Σ ⊆ C . We say that Σ is not redundant if � � ∀ σ ∈ Σ , τ � τ. τ ∈ Σ −{ σ } τ ∈ Σ 18

Non-redundancy An ingredient in understanding the role of cirquits for the nullity of subsets of E . Definition Let M be a matroid, and Σ ⊆ C . We say that Σ is not redundant if � � ∀ σ ∈ Σ , τ � τ. τ ∈ Σ −{ σ } τ ∈ Σ Definition Let M be a matroid on the ground set E , and σ ⊆ E . The degree of non-redundancy of σ is � deg ( σ ) = Max { #Σ | Σ ♥♦♥✲r❡❞✉♥❞❛♥t ❛♥❞ τ ⊆ σ } . τ ∈ Σ We have n ( σ ) = deg ( σ ) . 18

n ( σ ) � deg ( σ ) 19

n ( σ ) � deg ( σ ) 19

n ( σ ) � deg ( σ ) 19

n ( σ ) � deg ( σ ) 19

n ( σ ) � deg ( σ ) 19

n ( σ ) � deg ( σ ) 19

n ( σ ) � deg ( σ ) Important elements in proof: Proposition Let M be a matroid and X , Y ⊆ E . Then n ( X ∪ Y ) + n ( X ∩ Y ) � n ( X ) + n ( Y ) 20

n ( σ ) � deg ( σ ) Important elements in proof: Proposition Let M be a matroid and X , Y ⊆ E . Then n ( X ∪ Y ) + n ( X ∩ Y ) � n ( X ) + n ( Y ) Corollary If Σ ⊂ C is non redundant, then � � � n τ � #Σ . τ ∈ Σ 20

n ( σ ) � deg ( σ ) Important elements in proof: Proposition Let M be a matroid and X , Y ⊆ E . Then n ( X ∪ Y ) + n ( X ∩ Y ) � n ( X ) + n ( Y ) Corollary If Σ ⊂ C is non redundant, then � � � n τ � #Σ . τ ∈ Σ       � � � � �  ∩ σ  . n τ � n τ  + n ( σ ) − n τ    τ ∈ Σ τ ∈ Σ −{ σ } τ ∈ Σ −{ σ } 20

Content 1 Matroids - Hamming weights - Betti numbers Relation between the nullity function and 2 non-redundancy of circuits 3 Betti numbers and generalized Hamming weights Dualities 4 Examples and preliminary summary 5 Constant weight codes 6 Weight enumerators of matroids 7 Algebraic geometric codes 8 21

✐s ♠✐♥✐♠❛❧ ✇✐t❤ When do we have β i ,σ � = 0 ? Hochster and Björner: β i ,σ � = 0 ⇒ i = n ( σ ) . 22

✐s ♠✐♥✐♠❛❧ ✇✐t❤ When do we have β i ,σ � = 0 ? Hochster and Björner: β i ,σ � = 0 ⇒ i = n ( σ ) . Moreover, β n ( σ ) ,σ = ( − 1 ) r ( σ ) − 1 χ ( M | σ ) . 22

When do we have β i ,σ � = 0 ? Hochster and Björner: β i ,σ � = 0 ⇒ i = n ( σ ) . Moreover, β n ( σ ) ,σ = ( − 1 ) r ( σ ) − 1 χ ( M | σ ) . Theorem Let M be a matroid on the ground set E , and let σ ⊆ E . Then β i ,σ � = 0 ⇔ σ ✐s ♠✐♥✐♠❛❧ ✇✐t❤ n ( σ ) = i . 22

The Betti numbers decide the weight hierarchy [J-V]: Theorem Let M be a matroid on the ground set E of rank r . Then the generalized Hamming weights are given by d i = min { d | β i , d � = 0 } ❢♦r 1 � i � # E − r . 23

The Betti numbers decide the weight hierarchy [J-V]: Theorem Let M be a matroid on the ground set E of rank r . Then the generalized Hamming weights are given by d i = min { d | β i , d � = 0 } ❢♦r 1 � i � # E − r . Example Let C = {{ 1 , 2 , 3 , 4 } , { 1 , 4 , 5 } , { 1 , 6 } , { 2 , 3 , 4 , 6 } , { 2 , 3 , 5 } , { 4 , 5 , 6 }} . The Betti table is  1 0 0  3 2 0   2 7 4 2 23

The Betti numbers decide the weight hierarchy [J-V]: Theorem Let M be a matroid on the ground set E of rank r . Then the generalized Hamming weights are given by d i = min { d | β i , d � = 0 } ❢♦r 1 � i � # E − r . Example Let C = {{ 1 , 2 , 3 , 4 } , { 1 , 4 , 5 } , { 1 , 6 } , { 2 , 3 , 4 , 6 } , { 2 , 3 , 5 } , { 4 , 5 , 6 }} . The Betti table is  1 0 0  3 2 0   2 7 4 2 The weight hierarchy is therefore 2 , 4 , 6 23

MDS-codes A linear [ n , k ] -code satisfies d ≤ r + 1 = n − k + 1. If equality, the code is called MDS ( n − k is pr. def. the redundancy r of the code). Such codes correspond to uniform matroids U ( r , n ) . 24

MDS-codes A linear [ n , k ] -code satisfies d ≤ r + 1 = n − k + 1. If equality, the code is called MDS ( n − k is pr. def. the redundancy r of the code). Such codes correspond to uniform matroids U ( r , n ) . The resolution of the uniform matroid U ( r , n ) is: − S ( − ( r + 1 ))( r r )( n r + 1 ) 0 ← − R U ( r , n ) ← − S ← − S ( − ( r + 2 ))( r + 1 r )( n − S ( − ( r + 3 ))( r + 2 r )( n r + 2 ) ← r + 3 ) ← − S ( − ( n − 1 ))( n − 2 r )( n n − 1 ) ← − S ( − n )( n − 1 r )( n n ) ← ← − . . . ← − 0 . 24

MDS-codes A linear [ n , k ] -code satisfies d ≤ r + 1 = n − k + 1. If equality, the code is called MDS ( n − k is pr. def. the redundancy r of the code). Such codes correspond to uniform matroids U ( r , n ) . The resolution of the uniform matroid U ( r , n ) is: − S ( − ( r + 1 ))( r r )( n r + 1 ) 0 ← − R U ( r , n ) ← − S ← − S ( − ( r + 2 ))( r + 1 r )( n − S ( − ( r + 3 ))( r + 2 r )( n r + 2 ) ← r + 3 ) ← − S ( − ( n − 1 ))( n − 2 r )( n n − 1 ) ← − S ( − n )( n − 1 r )( n n ) ← ← − . . . ← − 0 . and the Betti diagram is: 1 · · · s · · · n − r �� n �� n � r � r + s − 1 � n − 1 �� n � � � r · · · · · · r r + 1 r r + s r n 24

MDS-codes A linear [ n , k ] -code satisfies d ≤ r + 1 = n − k + 1. If equality, the code is called MDS ( n − k is pr. def. the redundancy r of the code). Such codes correspond to uniform matroids U ( r , n ) . The resolution of the uniform matroid U ( r , n ) is: − S ( − ( r + 1 ))( r r )( n r + 1 ) 0 ← − R U ( r , n ) ← − S ← − S ( − ( r + 2 ))( r + 1 r )( n − S ( − ( r + 3 ))( r + 2 r )( n r + 2 ) ← r + 3 ) ← − S ( − ( n − 1 ))( n − 2 r )( n n − 1 ) ← − S ( − n )( n − 1 r )( n n ) ← ← − . . . ← − 0 . and the Betti diagram is: 1 · · · s · · · n − r �� n �� n � r � r + s − 1 � n − 1 �� n � � � r · · · · · · r r + 1 r r + s r n Hence the weight hierarchy is { n − k + 1 , . . . , n − 1 , n } . 24

Some negative results — β i �⇒ d i 25

Some negative results — β i �⇒ d i — d i �⇒ β i , β i , d , β i ,σ 25

Content 1 Matroids - Hamming weights - Betti numbers Relation between the nullity function and 2 non-redundancy of circuits 3 Betti numbers and generalized Hamming weights Dualities 4 Examples and preliminary summary 5 Constant weight codes 6 Weight enumerators of matroids 7 Algebraic geometric codes 8 26

Dual of a matroid - Wei duality Definition Let M be a matroid. Its dual M has the same ground set, and its set of bases is the set of complements of bases of M . 27

Dual of a matroid - Wei duality Definition Let M be a matroid. Its dual M has the same ground set, and its set of bases is the set of complements of bases of M . Theorem Let M be a matroid on E of cardinality n with weight hierarchy d 1 < · · · < d s . Then the weight hierarchy of M is d ′ 1 < · · · < d ′ n − s 27

Dual of a matroid - Wei duality Definition Let M be a matroid. Its dual M has the same ground set, and its set of bases is the set of complements of bases of M . Theorem Let M be a matroid on E of cardinality n with weight hierarchy d 1 < · · · < d s . Then the weight hierarchy of M is d ′ 1 < · · · < d ′ n − s and is such that { d 1 , · · · , d s , n − d ′ 1 + 1 , · · · , n − d ′ n − s + 1 } = E . 27

Dual of a matroid - Wei duality Definition Let M be a matroid. Its dual M has the same ground set, and its set of bases is the set of complements of bases of M . Theorem Let M be a matroid on E of cardinality n with weight hierarchy d 1 < · · · < d s . Then the weight hierarchy of M is d ′ 1 < · · · < d ′ n − s and is such that { d 1 , · · · , d s , n − d ′ 1 + 1 , · · · , n − d ′ n − s + 1 } = E . Corollary The N -graded Betti numbers of M give the weight hierarchy of M . 27

Alexander dual of a matroid Definition Let ∆ be a simplicial complex, with set of faces F . Then its Alexander dual ∆ ⋆ has set of faces F ⋆ = { τ | τ �∈ F} . 28

Alexander dual of a matroid Definition Let ∆ be a simplicial complex, with set of faces F . Then its Alexander dual ∆ ⋆ has set of faces F ⋆ = { τ | τ �∈ F} . Eagon-Reiner: the Alexander dual of a matroid has a linear resolution. 28

Alexander dual of a matroid Definition Let ∆ be a simplicial complex, with set of faces F . Then its Alexander dual ∆ ⋆ has set of faces F ⋆ = { τ | τ �∈ F} . Eagon-Reiner: the Alexander dual of a matroid has a linear resolution. Example C = {{ 1 , 4 } , { 2 , 3 }} 28

Alexander dual of a matroid Definition Let ∆ be a simplicial complex, with set of faces F . Then its Alexander dual ∆ ⋆ has set of faces F ⋆ = { τ | τ �∈ F} . Eagon-Reiner: the Alexander dual of a matroid has a linear resolution. Example C = {{ 1 , 4 } , { 2 , 3 }} 28

Alexander dual of a matroid Definition Let ∆ be a simplicial complex, with set of faces F . Then its Alexander dual ∆ ⋆ has set of faces F ⋆ = { τ | τ �∈ F} . Eagon-Reiner: the Alexander dual of a matroid has a linear resolution. Example C = {{ 1 , 4 } , { 2 , 3 }} Betti table of the Alexander dual M ⋆ : � � 4 4 1 2 . 28

Alexander dual of a matroid Definition Let ∆ be a simplicial complex, with set of faces F . Then its Alexander dual ∆ ⋆ has set of faces F ⋆ = { τ | τ �∈ F} . Eagon-Reiner: the Alexander dual of a matroid has a linear resolution. Example C = {{ 1 , 4 } , { 2 , 3 }} Betti table of the Alexander dual M ⋆ : � � 4 4 1 2 . The N -graded Betti numbers of the Alexander dual don’t in general give the weight hierarchy of M . 28

Content 1 Matroids - Hamming weights - Betti numbers Relation between the nullity function and 2 non-redundancy of circuits 3 Betti numbers and generalized Hamming weights Dualities 4 Examples and preliminary summary 5 Constant weight codes 6 Weight enumerators of matroids 7 Algebraic geometric codes 8 29

h -MDS codes Definition A linear code C of length n and dimension k is h -MDS if d h = n − k + h 30

h -MDS codes Definition A linear code C of length n and dimension k is h -MDS if d h = n − k + h Corollary C is h -MDS if and only if the right part F h ← F h + 1 ← · · · ← F k of the resolution is linear, and M ( C ) has no isthmus. 30

h -MDS codes Definition A linear code C of length n and dimension k is h -MDS if d h = n − k + h Corollary C is h -MDS if and only if the right part F h ← F h + 1 ← · · · ← F k of the resolution is linear, and M ( C ) has no isthmus. Corollary If C is non-degenerate, then it is MDS if and only if the Alexander dual of M ( C ) is also a matroid. 30

An example from algebraic codes Let X be an algebraic curve over F q of genus g in P g − 1 embedded by the canonical divisor K . 31

An example from algebraic codes Let X be an algebraic curve over F q of genus g in P g − 1 embedded by the canonical divisor K . Take (all) n distinct F q -rational points P 1 , · · · , P n and define a [ n × g ] matrix H where each column is a representative of P i . Let D = P 1 + · · · P n . 31

An example from algebraic codes Let X be an algebraic curve over F q of genus g in P g − 1 embedded by the canonical divisor K . Take (all) n distinct F q -rational points P 1 , · · · , P n and define a [ n × g ] matrix H where each column is a representative of P i . Let D = P 1 + · · · P n . Let M be the matroid associated to this matrix on the ground set { 1 , · · · , n } . 31

An example from algebraic codes Let X be an algebraic curve over F q of genus g in P g − 1 embedded by the canonical divisor K . Take (all) n distinct F q -rational points P 1 , · · · , P n and define a [ n × g ] matrix H where each column is a representative of P i . Let D = P 1 + · · · P n . Let M be the matroid associated to this matrix on the ground set { 1 , · · · , n } . A ⊆ E corresponds to a subdivisor A = P i 1 + · · · + P i s . 31

An example from algebraic codes Let X be an algebraic curve over F q of genus g in P g − 1 embedded by the canonical divisor K . Take (all) n distinct F q -rational points P 1 , · · · , P n and define a [ n × g ] matrix H where each column is a representative of P i . Let D = P 1 + · · · P n . Let M be the matroid associated to this matrix on the ground set { 1 , · · · , n } . A ⊆ E corresponds to a subdivisor A = P i 1 + · · · + P i s . Riemann-Roch : r ( A ) = l ( K ) − l ( K − A ) ⇒ n ( A ) = l ( A ) − 1 . Here r and n are matroid, and l is R.R. notation. 31

An example from algebraic codes Let X be an algebraic curve over F q of genus g in P g − 1 embedded by the canonical divisor K . Take (all) n distinct F q -rational points P 1 , · · · , P n and define a [ n × g ] matrix H where each column is a representative of P i . Let D = P 1 + · · · P n . Let M be the matroid associated to this matrix on the ground set { 1 , · · · , n } . A ⊆ E corresponds to a subdivisor A = P i 1 + · · · + P i s . Riemann-Roch : r ( A ) = l ( K ) − l ( K − A ) ⇒ n ( A ) = l ( A ) − 1 . Here r and n are matroid, and l is R.R. notation. A "quasi- t -gonality" disregarding divisors with repeated points: t D := min { deg A | l ( A ) = t + 1 } = d t = min { j | β j , t � = 0 } . 31