Codes, metrics, and applications Alexander Barg University of Maryland IITP RAS, Moscow ISIT 2016 Problems: § Distance distribution of codes § Duality of linear codes § Ordered linear codes and matroid invariants § Channel models matched to the ordered metrics; applications to polar codes, wiretap channels § Association schemes and bounds on the size of codes § Further extensions Acknowledgment: NSF grants CCF1217245 “Ordered metrics and their applications” CCF1422955, CCF0916919, CCF0807411 Talk summary Coding theory in a class of metric spaces: combinatorial and information-theoretic results and applications. § Channel models matched to the ordered metrics; applications to polar codes, wiretap channels § Association schemes and bounds on the size of codes § Further extensions Acknowledgment: NSF grants CCF1217245 “Ordered metrics and their applications” CCF1422955, CCF0916919, CCF0807411 Talk summary Coding theory in a class of metric spaces: combinatorial and information-theoretic results and applications. Problems: § Distance distribution of codes § Duality of linear codes § Ordered linear codes and matroid invariants Acknowledgment: NSF grants CCF1217245 “Ordered metrics and their applications” CCF1422955, CCF0916919, CCF0807411 Talk summary Coding theory in a class of metric spaces: combinatorial and information-theoretic results and applications. Problems: § Distance distribution of codes § Duality of linear codes § Ordered linear codes and matroid invariants § Channel models matched to the ordered metrics; applications to polar codes, wiretap channels § Association schemes and bounds on the size of codes § Further extensions Acknowledgment: NSF grants CCF1217245 “Ordered metrics and their applications” CCF1422955, CCF0916919, CCF0807411 Talk summary Coding theory in a class of metric spaces: combinatorial and information-theoretic results and applications. Problems: § Distance distribution of codes § Duality of linear codes § Ordered linear codes and matroid invariants § Channel models matched to the ordered metrics; applications to polar codes, wiretap channels § Association schemes and bounds on the size of codes § Further extensions Collaborators: Punarbasu Purkayastha, Woomyoung Park, Marcelo Firer, Maksim Skriganov, Min Ye, Talha Gulcu Talk summary Coding theory in a class of metric spaces: combinatorial and information-theoretic results and applications. Problems: § Distance distribution of codes § Duality of linear codes § Ordered linear codes and matroid invariants § Channel models matched to the ordered metrics; applications to polar codes, wiretap channels § Association schemes and bounds on the size of codes § Further extensions Acknowledgment: NSF grants CCF1217245 “Ordered metrics and their applications” CCF1422955, CCF0916919, CCF0807411 Outline § I. Brief recap: Linear codes, Weight distributions and duality, orthogonal arrays § II. Applications of the ordered distance § Wireless § Reed-Solomon codes § Approximation theory § III. Results on linear ordered codes § Shape distributions and bounds on codes § Duality of linear codes for poset metrics § Channel models § Polar codes C(dual) n C Dual code “ ty P Fq : px; yq “ 0 @x P u C(dual) (dual) Weight distribution of : Bi ; i “ 0; 1;:::; n Weight enumerators: ř ř n n´i i n (dual) n´i i BCpx; yq “ i“0 Bix y ; BC(dual) px; yq “ i“0 Bi x y I. Linear codes C n A linear code Ă Fq G; H generator and parity-check matrices Weight distribution Bi; i “ 0; 1;:::; n, where Bi “ 7tx P C : wpxq “ iu C(dual) n C Dual code “ ty P Fq : px; yq “ 0 @x P u C(dual) (dual) Weight distribution of : Bi ; i “ 0; 1;:::; n Weight enumerators: ř ř n n´i i n (dual) n´i i BCpx; yq “ i“0 Bix y ; BC(dual) px; yq “ i“0 Bi x y I. Linear codes C n A linear code Ă Fq G; H generator and parity-check matrices Weight distribution Bi; i “ 0; 1;:::; n, where Bi “ 7tx P C : wpxq “ iu Weight distributions are useful for analyzing structural properties of codes; probability of error under MAP or incomplete decoding I. Linear codes C n A linear code Ă Fq G; H generator and parity-check matrices Weight distribution Bi; i “ 0; 1;:::; n, where Bi “ 7tx P C : wpxq “ iu C(dual) n C Dual code “ ty P Fq : px; yq “ 0 @x P u C(dual) (dual) Weight distribution of : Bi ; i “ 0; 1;:::; n Weight enumerators: ř ř n n´i i n (dual) n´i i BCpx; yq “ i“0 Bix y ; BC(dual) px; yq “ i“0 Bi x y Approach via Fourier analysis: 1 ¸n B(dual) “ B K piq; j “ 0; 1;:::; n j |C| i j i“0 where ˆ ˙ˆ ˙ ¸i i n ´ i K piq “ p´iq` pq ´ 1qj´` j ` j ´ ` `“0 is a Krawtchouk polynomial Linear codes and duality The MacWilliams Theorem: 1 BCpx; yq “ BC(dual) px ` pq ´ 1qy; x ´ yq |C(dual)| Approach via Fourier analysis: 1 ¸n B(dual) “ B K piq; j “ 0; 1;:::; n j |C| i j i“0 where ˆ ˙ˆ ˙ ¸i i n ´ i K piq “ p´iq` pq ´ 1qj´` j ` j ´ ` `“0 is a Krawtchouk polynomial Linear codes and duality The MacWilliams Theorem: 1 BCpx; yq “ BC(dual) px ` pq ´ 1qy; x ´ yq |C(dual)| One of the basic facts in coding theory. Used for: § Classification of codes over various domains § Estimates of error probability § Bounds on the size of codes in terms of distance § Extensions used in sphere packing, optimality of lattices Linear codes and duality The MacWilliams Theorem: 1 BCpx; yq “ BC(dual) px ` pq ´ 1qy; x ´ yq |C(dual)| Approach via Fourier analysis: 1 ¸n B(dual) “ B K piq; j “ 0; 1;:::; n j |C| i j i“0 where ˆ ˙ˆ ˙ ¸i i n ´ i K piq “ p´iq` pq ´ 1qj´` j ` j ´ ` `“0 is a Krawtchouk polynomial § ´ ¯ k n´k qy x ´ y BCpx; yq “ px ´ yq y ZC ; (Greene 1976) x ´ y y § This connection extends to higher support weights (B ’97) (Wei ’91, Ozarow-Wyner ’84). § Codes and matroids: If the code is considered as an Fq-representation of a matroid M on the set t1; 2;:::; nu; then ZCpx; yq is the Whitney function of M Linear codes and duality The MacWilliams Theorem: § Linear algebraic approach: C Let A Ă t1; 2;:::; řnu; ρA “ rankpGpAqq; k “ dim k´ρA |A|´ρA Define ZCpx; yq “ AĂrns x y : Then ZCpx; yq “ ZC(dual) py; xq ZCpx; yq is the Whitney rank-nullity function of C § This connection extends to higher support weights (B ’97) (Wei ’91, Ozarow-Wyner ’84). § Codes and matroids: If the code is considered as an Fq-representation of a matroid M on the set t1; 2;:::; nu; then ZCpx; yq is the Whitney function of M Linear codes and duality The MacWilliams Theorem: § Linear algebraic approach: C Let A Ă t1; 2;:::; řnu; ρA “ rankpGpAqq; k “ dim k´ρA |A|´ρA Define ZCpx; yq “ AĂrns x y : Then ZCpx; yq “ ZC(dual) py; xq ZCpx; yq is the Whitney rank-nullity function of C § ´ ¯ k n´k qy x ´ y BCpx; yq “ px ´ yq y ZC ; (Greene 1976) x ´ y y § Codes and matroids: If the code is considered as an Fq-representation of a matroid M on the set t1; 2;:::; nu; then ZCpx; yq is the Whitney function of M Linear codes and duality The MacWilliams Theorem: § Linear algebraic approach: C Let A Ă t1; 2;:::; řnu; ρA “ rankpGpAqq; k “ dim k´ρA |A|´ρA Define ZCpx; yq “ AĂrns x y : Then ZCpx; yq “ ZC(dual) py; xq ZCpx; yq is the Whitney rank-nullity function of C § ´ ¯ k n´k qy x ´ y BCpx; yq “ px ´ yq y ZC ; (Greene 1976) x ´ y y § This connection extends to higher support weights (B ’97) (Wei ’91, Ozarow-Wyner ’84). Linear codes and duality The MacWilliams Theorem: § Linear algebraic approach: C Let A Ă t1; 2;:::; řnu; ρA “ rankpGpAqq; k “ dim k´ρA |A|´ρA Define ZCpx; yq “ AĂrns x y : Then ZCpx; yq “ ZC(dual) py; xq ZCpx; yq is the Whitney rank-nullity function of C § ´ ¯ k n´k qy x ´ y BCpx; yq “ px ´ yq y ZC ; (Greene 1976) x ´ y y § This connection extends to higher support weights (B ’97) (Wei ’91, Ozarow-Wyner ’84). § Codes and matroids: If the code is considered as an Fq-representation of a matroid M on the set t1; 2;:::; nu; then ZCpx; yq is the Whitney function of M 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 OAp8; 4; 1; 3q : 0 1 1 1 1 0 1 1 1 1 0 1 1 1 1 0 OAs form an example of designs in association schemes (Delsarte ’73) Orthogonal arrays Consider a code C, and suppose that C(dual) (dual) dp q “ t ` 1; i.e., Bi “ 0; i “ 1; 2;:::; t Then C is called an orthogonal array of strength t (C. R. Rao, 1946+) 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 OAp8; 4; 1; 3q : 0 1 1 1 1 0 1 1 1 1 0 1 1 1 1 0 OAs form an example of designs in association schemes (Delsarte ’73) Orthogonal arrays Consider a code C, and suppose that C(dual) (dual) dp q “ t ` 1; i.e., Bi “ 0; i “ 1; 2;:::; t Then C is called an orthogonal array of strength t (C. R. Rao, 1946+) 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 OAp8; 4; 1; 3q : 0 1 1 1 1 0 1 1 1 1 0 1 1 1 1 0 Orthogonal arrays Consider a code C, and suppose that C(dual) (dual) dp q “ t ` 1; i.e., Bi “ 0; i “ 1; 2;:::; t Then C is called an orthogonal array of strength t (C. R. Rao, 1946+) 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 OAp8; 4; 1; 3q : 0 1 1 1 1 0 1 1 1 1 0 1 1 1 1 0 OAs form an example of designs in association schemes (Delsarte ’73) § Hamming distance § Lee distance § Levenshtein (edit) distance § `1 distance; Kendall tau metric; Chebyshev (`8) distance § Subspace distance § Ordered metrics (Niederreiter ’92; Brualdi et al., ’95; Rosenbloom-Tsfasman, ’97) M.Deza and E.