I am a friend of Finite Geometry! On the q-Analogue of Cauchy Matrices Alessandro Neri 17-21 June 2019 - VUB Alessandro Neri 19 June 2019 1 / 19 On the q-Analogue of Cauchy Matrices Alessandro Neri I am a friend of Finite Geometry! 17-21 June 2019 - VUB Alessandro Neri 19 June 2019 1 / 19 q-Analogues Model Finite set Finite dim vector space over Fq Element 1-dim subspace ; f0g Cardinality Dimension Intersection Intersection Union Sum Alessandro Neri 19 June 2019 1 / 19 Examples 1. Binomials and q-binomials: k−1 k−1 n Y n − i n Y 1 − qn−i = ; = : k k − i k 1 − qk−i i=0 q i=0 2. (Chu)-Vandermonde and q-Vandermonde identity: k k m + n X m n m + n X m n = ; = qj(m−k+j): k j k − j k k − j j j=0 q j=0 q q 3. Polynomials and q-polynomials: k q qk a0 + a1x + ::: + ak x ; a0x + a1x + ::: + ak x : 4. Gamma and q-Gamma functions: 1 − qx Γ(z + 1) = zΓ(z); Γ (x + 1) = Γ (x): q 1 − q q Alessandro Neri 19 June 2019 2 / 19 For n = k, det(V ) 6= 0 if and only if the αi 's are all distinct. jfα1; : : : ; αngj = n: In particular, all the k × k minors of V are non-zero. Vandermonde Matrix Let k ≤ n. 0 1 1 ::: 1 1 B α1 α2 : : : αn C B C B α2 α2 : : : α2 C V = B 1 2 n C B . C @ . A k−1 k−1 k−1 α1 α2 : : : αn Alessandro Neri 19 June 2019 3 / 19 Vandermonde Matrix Let k ≤ n. 0 1 1 ::: 1 1 B α1 α2 : : : αn C B C B α2 α2 : : : α2 C V = B 1 2 n C B . C @ . A k−1 k−1 k−1 α1 α2 : : : αn For n = k, det(V ) 6= 0 if and only if the αi 's are all distinct. jfα1; : : : ; αngj = n: In particular, all the k × k minors of V are non-zero. Alessandro Neri 19 June 2019 3 / 19 For n = k, det(G) 6= 0 if and only if the gi 's are all Fq-linearly independent. dimFq hg1;:::; gniFq = n: In particular, for every M 2 GLn(Fq), all the k × k minors of GM are non-zero. Moore Matrix Let k ≤ n. 0 g1 g2 ::: gn 1 q q q B g1 g2 ::: gn C B C G = B . C ; @ . A qk−1 qk−1 qk−1 g1 g2 ::: gn Alessandro Neri 19 June 2019 4 / 19 Moore Matrix Let k ≤ n. 0 g1 g2 ::: gn 1 q q q B g1 g2 ::: gn C B C G = B . C ; @ . A qk−1 qk−1 qk−1 g1 g2 ::: gn For n = k, det(G) 6= 0 if and only if the gi 's are all Fq-linearly independent. dimFq hg1;:::; gniFq = n: In particular, for every M 2 GLn(Fq), all the k × k minors of GM are non-zero. Alessandro Neri 19 June 2019 4 / 19 Moore Matrix Let k ≤ n, σ : x 7−! x q 0 1 g1 g2 ::: gn B σ(g1) σ(g2) : : : σ(gn) C G B C ; = B . C @ . A k−1 k−1 k−1 σ (g1) σ (g2) : : : σ (gn) For n = k, det(G) 6= 0 if and only if the gi 's are all Fq-linearly independent. dimFq hg1;:::; gniFq = n: In particular, for every M 2 GLn(Fq), all the k × k minors of GM are non-zero. Alessandro Neri 19 June 2019 5 / 19 Moore matrix 1. Dickson used the Moore matrix for finding the modular invariants of the general linear group over a finite field. 2. It is widely used in the study of normal bases of finite fields. 3. Y Y det(G) = (c1g1 + ··· + ci−1gi−1 + gi ) : 1≤i≤n c1;:::;ci−12Fq Alessandro Neri 19 June 2019 6 / 19 r×s The matrix X 2 Fq defined by ci dj Xi;j = xi − yj is called Generalized Cauchy (GC) Matrix. For j ≤ minfr; sg, all the j × j minors are non-zero. Generalized Cauchy Matrix (1) x1;:::; xr 2 Fq pairwise distinct, (2) y1;:::; ys 2 Fq pairwise distinct, (3) y1;:::; ys 2 Fq n fx1;:::; xr g, ∗ (4) c1;:::; cr ; d1;:::; ds 2 Fq. Alessandro Neri 19 June 2019 7 / 19 Generalized Cauchy Matrix (1) x1;:::; xr 2 Fq pairwise distinct, (2) y1;:::; ys 2 Fq pairwise distinct, (3) y1;:::; ys 2 Fq n fx1;:::; xr g, ∗ (4) c1;:::; cr ; d1;:::; ds 2 Fq. r×s The matrix X 2 Fq defined by ci dj Xi;j = xi − yj is called Generalized Cauchy (GC) Matrix. For j ≤ minfr; sg, all the j × j minors are non-zero. Alessandro Neri 19 June 2019 7 / 19 Question How to define a q-analogue of Cauchy matrices? Alessandro Neri 19 June 2019 8 / 19 α1; : : : ; αn 2 Fq distinct elements ∗ b1;:::; bn 2 Fq C = f(b1f (α1); b2f (α2);:::; bnf (αn)) j f 2 Fq[x]<k g THEN C is the Generalized Reed-Solomon code GRSn;k (α; b) Generalized Reed-Solomon Codes [Reed, Solomon '60] Fq[x]<k := ff (x) 2 Fq[x] j deg f < kg : Alessandro Neri 19 June 2019 9 / 19 THEN C is the Generalized Reed-Solomon code GRSn;k (α; b) Generalized Reed-Solomon Codes [Reed, Solomon '60] Fq[x]<k := ff (x) 2 Fq[x] j deg f < kg : α1; : : : ; αn 2 Fq distinct elements ∗ b1;:::; bn 2 Fq C = f(b1f (α1); b2f (α2);:::; bnf (αn)) j f 2 Fq[x]<k g Alessandro Neri 19 June 2019 9 / 19 Generalized Reed-Solomon Codes [Reed, Solomon '60] Fq[x]<k := ff (x) 2 Fq[x] j deg f < kg : α1; : : : ; αn 2 Fq distinct elements ∗ b1;:::; bn 2 Fq C = f(b1f (α1); b2f (α2);:::; bnf (αn)) j f 2 Fq[x]<k g THEN C is the Generalized Reed-Solomon code GRSn;k (α; b) Alessandro Neri 19 June 2019 9 / 19 Generalized Reed-Solomon Codes [Reed, Solomon '60] Fq[x]<k := ff (x) 2 Fq[x] j deg f < kg : α1; : : : ; αn 2 Fq s.t. jfα1; : : : ; αngj = n ∗ b1;:::; bn 2 Fq C = f(b1f (α1); b2f (α2);:::; bnf (αn)) j f 2 Fq[x]<k g THEN C is the Generalized Reed-Solomon code GRSn;k (α; b) Alessandro Neri 19 June 2019 9 / 19 Canonical Generator Matrix for GRS codes Consider the canonical monomial basis f1; x;:::; x k−1g and evaluate it. We get the following generator matrix Weighted Vandermonde (WV) matrix 0 1 1 ::: 1 1 B α1 α2 : : : αn C B C B α2 α2 : : : α2 C B 1 2 n C diag(b) B . C @ . A k−1 k−1 k−1 α1 α2 : : : αn Alessandro Neri 19 June 2019 10 / 19 3. If C is MDS the generator matrix in RREF is of the form (Ik j X ). 4. GRS codes are MDS. Theorem [Roth, Seroussi '85] There is a 1-1 correspondence between GC matrices and GRS codes codes given by X ! rowsp( Ik j X ): GRS Codes and Cauchy Matrices k×n 1. Every [n; k]q code has many generator matrices G 2 Fq . 2. Every [n; k]q code has a unique generator matrix in Reduced Row Echelon Form (RREF). Alessandro Neri 19 June 2019 11 / 19 Theorem [Roth, Seroussi '85] There is a 1-1 correspondence between GC matrices and GRS codes codes given by X ! rowsp( Ik j X ): GRS Codes and Cauchy Matrices k×n 1. Every [n; k]q code has many generator matrices G 2 Fq . 2. Every [n; k]q code has a unique generator matrix in Reduced Row Echelon Form (RREF). 3. If C is MDS the generator matrix in RREF is of the form (Ik j X ). 4. GRS codes are MDS. Alessandro Neri 19 June 2019 11 / 19 Theorem [Roth, Seroussi '85] There is a 1-1 correspondence between GC matrices and GRS codes codes given by X ! rowsp( Ik j X ): GRS Codes and Cauchy Matrices k×n 1. Every [n; k]q code has many generator matrices G 2 Fq . 2. Every [n; k]q code has a unique generator matrix in Reduced Row Echelon Form (RREF). 3. If C is MDS the generator matrix in RREF is of the form (Ik j X ). 4. GRS codes are MDS. Alessandro Neri 19 June 2019 11 / 19 GRS Codes and Cauchy Matrices k×n 1. Every [n; k]q code has many generator matrices G 2 Fq . 2. Every [n; k]q code has a unique generator matrix in Reduced Row Echelon Form (RREF). 3. If C is MDS the generator matrix in RREF is of the form (Ik j X ). 4. GRS codes are MDS. Theorem [Roth, Seroussi '85] There is a 1-1 correspondence between GC matrices and GRS codes codes given by X ! rowsp( Ik j X ): Alessandro Neri 19 June 2019 11 / 19 Rank Distance m×n The rank distance dR on Fq is defined by m×n dR (X ; Y ) := rk(X − Y ); X ; Y 2 Fq : n The rank distance dR on Fqm is defined by dR (u; v) := dimhu1 − v1; u2 − v2;:::; un − vniFq : n A rank metric code C is a subset of Fqm equipped with the rank distance. General Setting Fqm extension field of degree m of a finite field Fq. ∼ m $ Fqm = Fq as vector spaces over Fq. n ∼ m×n $ Fqm = Fq as vector spaces over Fq.