Mac Williams identities for linear codes as Riemann-Roch conditions Azniv Kasparian, Ivan Marinov 1

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1 Mac Williams identities for linear codes as Riemann-Roch conditions Azniv Kasparian, Ivan Marinov 1 1 Partially supported by Contract 57/ with the Scientific Foundation of Kliment Ohridski University of Sofia.

2 The genus of a linear code Let C be an F q -linear [n, k, d]-code. The genus of C is the deviation g := n + 1 k d from the equality in the Singleton bound n + 1 k d 0. Let us{ denote by g = k + 1 d the genus of the dual code C = a F n q a, c = n } a i c i = 0 for c C. i=1

3 The genus of a linear code Let C be an F q -linear [n, k, d]-code. The genus of C is the deviation g := n + 1 k d from the equality in the Singleton bound n + 1 k d 0. Let us{ denote by g = k + 1 d the genus of the dual code C = a F n q a, c = n } a i c i = 0 for c C. i=1

4 The homogeneous weight enumerator of a linear code If W (w) C is the number of the words c C of weight 1 w n then W C (x, y) = x n + n w=d homogeneous weight enumerator of C. Denote by M n,s (x, y) = x n + n M (w) n,s = ( n w ) w s W (w) C xn w y w is called the w=s M (w) n,s x n w y w with ( 1) i( ) w i (q w+1 s i 1) the homogeneous weight enumerator of an MDS-code with parameters [n, n + 1 s, s].

5 The homogeneous weight enumerator of a linear code If W (w) C is the number of the words c C of weight 1 w n then W C (x, y) = x n + n w=d homogeneous weight enumerator of C. Denote by M n,s (x, y) = x n + n M (w) n,s = ( n w ) w s W (w) C xn w y w is called the w=s M (w) n,s x n w y w with ( 1) i( ) w i (q w+1 s i 1) the homogeneous weight enumerator of an MDS-code with parameters [n, n + 1 s, s].

6 The ζ-polynomial and the ζ-function of a linear code Theorem (Duursma ): For any linear code C of genus g 0 with dual C of genus g 0 there is a unique g+g ζ-polynomial P C (t) = a i t i Q[t] with W C (x, y) = g+g a i M n,d+i (x, y) and P C (1) = 1. The quotient ζ C (t) = P C(t) (1 t)(1 qt) is the ζ-function of C.

7 The ζ-polynomial and the ζ-function of a linear code Theorem (Duursma ): For any linear code C of genus g 0 with dual C of genus g 0 there is a unique g+g ζ-polynomial P C (t) = a i t i Q[t] with W C (x, y) = g+g a i M n,d+i (x, y) and P C (1) = 1. The quotient ζ C (t) = P C(t) (1 t)(1 qt) is the ζ-function of C.

8 Algebro-geometric Goppa codes Let X/F q P N (F q ) be a smooth irreducible curve of genus g, P 1,..., P n X(F q ) = X P N (F q ), D = P P n and G 1,..., G h be a complete set of representatives of the linear equivalence classes of the divisors of F q (X) of degree 2g 2 < m < n with Supp(G i ) Supp(D) = for 1 i h.

9 Algebro-geometric Goppa codes Let X/F q P N (F q ) be a smooth irreducible curve of genus g, P 1,..., P n X(F q ) = X P N (F q ), D = P P n and G 1,..., G h be a complete set of representatives of the linear equivalence classes of the divisors of F q (X) of degree 2g 2 < m < n with Supp(G i ) Supp(D) = for 1 i h.

10 Algebro-geometric Goppa codes Let X/F q P N (F q ) be a smooth irreducible curve of genus g, P 1,..., P n X(F q ) = X P N (F q ), D = P P n and G 1,..., G h be a complete set of representatives of the linear equivalence classes of the divisors of F q (X) of degree 2g 2 < m < n with Supp(G i ) Supp(D) = for 1 i h.

11 Algebro-geometric Goppa codes Let X/F q P N (F q ) be a smooth irreducible curve of genus g, P 1,..., P n X(F q ) = X P N (F q ), D = P P n and G 1,..., G h be a complete set of representatives of the linear equivalence classes of the divisors of F q (X) of degree 2g 2 < m < n with Supp(G i ) Supp(D) = for 1 i h.

12 Algebro-geometric Goppa codes Let X/F q P N (F q ) be a smooth irreducible curve of genus g, P 1,..., P n X(F q ) = X P N (F q ), D = P P n and G 1,..., G h be a complete set of representatives of the linear equivalence classes of the divisors of F q (X) of degree 2g 2 < m < n with Supp(G i ) Supp(D) = for 1 i h.

13 Algebro-geometric Goppa codes The evaluation maps E D : H 0 (X, O X ([G i ])) F n q, E D (f) = (f(p 1 ),..., f(p n )) of the global sections f of the line bundles on X, associated with G i are F q -linear. Their images C i = E D H 0 (X, O X ([G i ])) F n q are F q -linear codes of genus g i g, known as algebro-geometric Goppa codes.

14 Algebro-geometric Goppa codes The evaluation maps E D : H 0 (X, O X ([G i ])) F n q, E D (f) = (f(p 1 ),..., f(p n )) of the global sections f of the line bundles on X, associated with G i are F q -linear. Their images C i = E D H 0 (X, O X ([G i ])) F n q are F q -linear codes of genus g i g, known as algebro-geometric Goppa codes.

15 The ζ-functions of X and C i If X(F q r) is the number of the F q r-rational points X(F q r) := X ( P N (F q r) of X then the formal power series ) ζ X (t) := exp X(F q r) tr r is called the ζ-function of X. r=1 Duursma s considerations imply that the ζ-functions of X and C i satisfy the equality ζ X (t) = h t g g iζ Ci (t). i=1

16 The ζ-functions of X and C i If X(F q r) is the number of the F q r-rational points X(F q r) := X ( P N (F q r) of X then the formal power series ) ζ X (t) := exp X(F q r) tr r is called the ζ-function of X. r=1 Duursma s considerations imply that the ζ-functions of X and C i satisfy the equality ζ X (t) = h t g g iζ Ci (t). i=1

17 Divisors on curves The absolute Galois group Gal(F q /F q ) acts on any smooth irreducible curve X/F q P N (F q ) with finite orbits and deg Orb Gal(Fq/F q) (x) := OrbGal(Fq/F q) (x). The Z-linear combinations D = a 1 ν a s ν s of Gal(F q /F q )-orbits ν j X are called divisors on X. The degree of D is deg D = a 1 deg ν a s deg ν s.

18 Divisors on curves The absolute Galois group Gal(F q /F q ) acts on any smooth irreducible curve X/F q P N (F q ) with finite orbits and deg Orb Gal(Fq/F q) (x) := OrbGal(Fq/F q) (x). The Z-linear combinations D = a 1 ν a s ν s of Gal(F q /F q )-orbits ν j X are called divisors on X. The degree of D is deg D = a 1 deg ν a s deg ν s.

19 Divisors on curves The absolute Galois group Gal(F q /F q ) acts on any smooth irreducible curve X/F q P N (F q ) with finite orbits and deg Orb Gal(Fq/F q) (x) := OrbGal(Fq/F q) (x). The Z-linear combinations D = a 1 ν a s ν s of Gal(F q /F q )-orbits ν j X are called divisors on X. The degree of D is deg D = a 1 deg ν a s deg ν s.

20 Effective divisors of fixed degree A divisor D = a 1 ν a s ν s is effective if all of its non-zero coefficients a j > 0 are positive. There are finitely many Gal(F q /F q )-orbits on X of fixed degree and, therefore, a finite number A m (X) Z 0 of effective divisors on X of degree m Z 0. The ζ-function of X is ζ X (t) = m=0 A m (X)t m.

21 Effective divisors of fixed degree A divisor D = a 1 ν a s ν s is effective if all of its non-zero coefficients a j > 0 are positive. There are finitely many Gal(F q /F q )-orbits on X of fixed degree and, therefore, a finite number A m (X) Z 0 of effective divisors on X of degree m Z 0. The ζ-function of X is ζ X (t) = m=0 A m (X)t m.

22 Effective divisors of fixed degree A divisor D = a 1 ν a s ν s is effective if all of its non-zero coefficients a j > 0 are positive. There are finitely many Gal(F q /F q )-orbits on X of fixed degree and, therefore, a finite number A m (X) Z 0 of effective divisors on X of degree m Z 0. The ζ-function of X is ζ X (t) = m=0 A m (X)t m.

23 Riemann-Roch Conditions for a curve Immediate consequences of the Riemann-Roch Theorem on a smooth irreducible curve X/F q P N (F q ) of genus g are the Riemann-Roch Conditions A m (X) = q m g+1 A 2g 2 m (X) + (q m g+1 1)Res 1 (ζ X (t)) for m g and the residuum Res 1 (ζ X (t)) of ζ X (t) at t = 1.

24 Polarized Riemann-Roch Conditions Definition: Formal power series ζ(t) = m=0 A m t m and ζ (t) = A mt m satisfy the Polarized Riemann-Roch Conditions PRRC(g, g ) for some g, g Z 0 if A m = q m g+1 A g+g 2 m +(qm g+1 1)Res 1 (ζ(t)) for m g, A g 1 = A g 1 and A m = q m g +1 A g+g 2 m+(q m g +1 1)Res 1 (ζ (t)) for m g, where Res 1 (ζ(t)), Res 1 (ζ (t)) are the residuums at t = 1.

25 Riemann-Roch Conditions imply rationality Note that PRRC(g, g ) imply the recurrence relations A m+2 (q + 1)A m+1 + qa m = A m+2 (q + 1)A m+1 + qa m = 0 for m g + g 1, which hold exactly when ζ(t) = P(t) (1 t)(1 qt), ζ (t) = for polynomials P(t), P (t) C[t]. P (t) (1 t)(1 qt)

26 Mac Williams identities as PRRC Theorem: Mac Williams identities for an F q -linear [n, k, d]-code C of genus g := n + 1 k d 0 and its dual C F n q of genus g = k + 1 d 0 are equivalent to the Polarized Riemann-Roch Conditions PRRC(g, g ) on their ζ-functions ζ C (t), ζ C (t).

27 Definition of Duursma s reduced polynomial Proposition (KM ): Let C be an F q -linear [n, k, d]-code of genus g = n + 1 k d 1, whose dual C is of genus g = k + 1 d 1. Then there is a unique Duursma s reduced g+g 2 polynomial D C (t) = c i t i Q[t], such that W C (x, y) = M n,n+1 k (x, y) + g+g 2 ( (q 1)c n i d+i) (x y) n d i y d+i.

28 D C and D C are determined by g + g 1 parameters Corollary: The lower parts ϕ C (t) = g 2 of Duursma s reduced polynomials D C (t) = D C (t) = g+g 2 c i t i, ϕ C (t) = g+g 2 c i t i, g 2 c i t i and the number c g 1 = c g 1 Q determine uniquely ( ) 1 D C (t) = ϕ C (t) + c g 1 t g 1 + ϕ C q g 1 t g+g 2, qt ( ) 1 D C (t) = ϕ C (t) + c g 1 t g 1 + ϕ C q g 1 t g+g 2. qt c i t i

29 The coefficients of Duursma s reduced polynomial Corollary: If C if an F q -linear code of genus g 1 with g+g 2 Duursma s reduced polynomial D C (t) = c i t i Q[t], then ( ) n c i Z 0 for 0 i g + g 2. d + i A linear code C F n q is non-degenerate if it is not contained in a coordinate hyperplane V(x i ) = {a F n q a i = 0} for some 1 i n.

30 The coefficients of Duursma s reduced polynomial Corollary: If C if an F q -linear code of genus g 1 with g+g 2 Duursma s reduced polynomial D C (t) = c i t i Q[t], then ( ) n c i Z 0 for 0 i g + g 2. d + i A linear code C F n q is non-degenerate if it is not contained in a coordinate hyperplane V(x i ) = {a F n q a i = 0} for some 1 i n.

31 An averaging interpretation of the coefficients of Duursma s reduced polynomial Proposition: Let C be a non-degenerate F q -linear code with g+g 2 Duursma s reduced polynomial D C (t) = c i t i Q[t] and P(C) ( β) = {[a] P(C) P(F n q) Supp([a]) β} for β = {β 1,..., β d+i } {1,..., n} with 0 i g 1. Then ( ) n 1 c i = P(C) ( β) d + i β={β 1,...,β d+i } {1,...,n} is the average cardinality of an intersection of P(C) with n d i coordinate hyperplanes in P(F n q).

32 Probabilistic interpretations of the coefficients of Duursma s reduced polynomial Proposition: Let C be an F q -linear code with Duursma s g+g 2 reduced polynomial D C (t) = c i t i Q[t]. If π (w) P(C), respectively, π (w) P(C ) is the probability of [b] P(Fn q) with wt([b]) = w to belong to P(C), respectively, to P(C ), then c i = d+i w=d π (w) P(C) [ n d i c i = q i g+1 ( d + i w w=d π (w) P(C ) ) (q 1) w 1 for 0 i g 1; ( ] n d i )(q 1) w 1, g i g+g 2. w

33 Probabilistic interpretations of the coefficients of Duursma s reduced polynomial Proposition: Let C be an F q -linear code with Duursma s g+g 2 reduced polynomial D C (t) = c i t i Q[t]. If π (w) [a] is the probability of β = {β 1,..., β w } {1,..., n} to contain the support Supp([a]) of [a] P(F n q), then c i = c i = q i g+1 [a] P(C) [b] P(C ) π (d+i) [a] for 0 i g 1; π (n d i) [b] for g i g + g 2.

34 Thank you for your attention!

The minimum distance of codes in an array coming from telescopic semigroups

The minimum distance of codes in an array coming from telescopic semigroups Christoph Kirfel and Ruud Pellikaan in IEEE Transactions Information Theory, vol. 41, pp. 1720 1732, (1995) Abstract The concept