Error-Correcting Codes over Rings Lecture 2: Cyclic Codes W. Cary Huffman Department of Mathematics and Statistics Loyola University Chicago [email protected] Noncommutative Rings and Their Applications Universit´ed'Atois Lens, France June 25, 2019 References • E. Prange, Technical Notes AFCRL - \Cyclic error-correcting codes in two symbols", TN-57-103 (September, 1957) - \Some cyclic error-correcting codes with simple decoding algorithms", TN-58-156 (April, 1958) - \The use of coset equivalence in the analysis and decoding of group codes," TN-59-164 (1959) - \An algorithm for factoring x n − 1 over a finite field”, TN-59-175 (October, 1959) • W. W. Peterson, Error-Correcting Codes, MIT Press, Cambridge, MA, 1961. Cyclic Codes - the Beginning People Eugene Prange (Air Force Cambridge Research Laboratory, Bedford, Massachusetts) and W. Wesley Peterson (IBM, MIT, U. of Florida, U. of Hawaii). • W. W. Peterson, Error-Correcting Codes, MIT Press, Cambridge, MA, 1961. Cyclic Codes - the Beginning People Eugene Prange (Air Force Cambridge Research Laboratory, Bedford, Massachusetts) and W. Wesley Peterson (IBM, MIT, U. of Florida, U. of Hawaii). References • E. Prange, Technical Notes AFCRL - \Cyclic error-correcting codes in two symbols", TN-57-103 (September, 1957) - \Some cyclic error-correcting codes with simple decoding algorithms", TN-58-156 (April, 1958) - \The use of coset equivalence in the analysis and decoding of group codes," TN-59-164 (1959) - \An algorithm for factoring x n − 1 over a finite field”, TN-59-175 (October, 1959) Cyclic Codes - the Beginning People Eugene Prange (Air Force Cambridge Research Laboratory, Bedford, Massachusetts) and W. Wesley Peterson (IBM, MIT, U. of Florida, U. of Hawaii). References • E. Prange, Technical Notes AFCRL - \Cyclic error-correcting codes in two symbols", TN-57-103 (September, 1957) - \Some cyclic error-correcting codes with simple decoding algorithms", TN-58-156 (April, 1958) - \The use of coset equivalence in the analysis and decoding of group codes," TN-59-164 (1959) - \An algorithm for factoring x n − 1 over a finite field”, TN-59-175 (October, 1959) • W. W. Peterson, Error-Correcting Codes, MIT Press, Cambridge, MA, 1961. Definition n Let C ⊆ A . C is cyclic provided for all c = c0c1 ··· cn−1 2 C, the 0 cyclic shift c = cn−1c0 ··· cn−2 2 C. Remark A cyclic code is closed under cyclic shifts, with wrap-around, of any amount in either direction. Definition of a Cyclic Code Notation n Subscript change: v = v0v1 ··· vn−1 2 A . Remark A cyclic code is closed under cyclic shifts, with wrap-around, of any amount in either direction. Definition of a Cyclic Code Notation n Subscript change: v = v0v1 ··· vn−1 2 A . Definition n Let C ⊆ A . C is cyclic provided for all c = c0c1 ··· cn−1 2 C, the 0 cyclic shift c = cn−1c0 ··· cn−2 2 C. Definition of a Cyclic Code Notation n Subscript change: v = v0v1 ··· vn−1 2 A . Definition n Let C ⊆ A . C is cyclic provided for all c = c0c1 ··· cn−1 2 C, the 0 cyclic shift c = cn−1c0 ··· cn−2 2 C. Remark A cyclic code is closed under cyclic shifts, with wrap-around, of any amount in either direction. • R[x] is the ring of polynomials in x with coefficients in R. • Let hxn − 1i denote the two-sided principal ideal of R[x] n n generated by x − 1. Let PR;n = R[x]=hx − 1i. n n • Define ι : R !PR;n as follows: If c = c0c1 ··· cn−1 2 R , let n−1 n ι(c) = c(x)= c0 + c1x + ··· + cn−1x + hx − 1i. Remarks n • Both R and PR;n are left (or right) R-modules under addition and left (or right) scalar multiplication by elements of n R. The map ι is an R-module isomorphism of R onto PR;n. • Images under ι of left-linear (or right-linear) codes in Rn are left (or right) R-submodules of PR;n. Polynomial Setting Notation Let R be a finite ring with unity. Let x be an indeterminate over R and n a positive integer. • Let hxn − 1i denote the two-sided principal ideal of R[x] n n generated by x − 1. Let PR;n = R[x]=hx − 1i. n n • Define ι : R !PR;n as follows: If c = c0c1 ··· cn−1 2 R , let n−1 n ι(c) = c(x)= c0 + c1x + ··· + cn−1x + hx − 1i. Remarks n • Both R and PR;n are left (or right) R-modules under addition and left (or right) scalar multiplication by elements of n R. The map ι is an R-module isomorphism of R onto PR;n. • Images under ι of left-linear (or right-linear) codes in Rn are left (or right) R-submodules of PR;n. Polynomial Setting Notation Let R be a finite ring with unity. Let x be an indeterminate over R and n a positive integer. • R[x] is the ring of polynomials in x with coefficients in R. n n • Define ι : R !PR;n as follows: If c = c0c1 ··· cn−1 2 R , let n−1 n ι(c) = c(x)= c0 + c1x + ··· + cn−1x + hx − 1i. Remarks n • Both R and PR;n are left (or right) R-modules under addition and left (or right) scalar multiplication by elements of n R. The map ι is an R-module isomorphism of R onto PR;n. • Images under ι of left-linear (or right-linear) codes in Rn are left (or right) R-submodules of PR;n. Polynomial Setting Notation Let R be a finite ring with unity. Let x be an indeterminate over R and n a positive integer. • R[x] is the ring of polynomials in x with coefficients in R. • Let hxn − 1i denote the two-sided principal ideal of R[x] n n generated by x − 1. Let PR;n = R[x]=hx − 1i. Remarks n • Both R and PR;n are left (or right) R-modules under addition and left (or right) scalar multiplication by elements of n R. The map ι is an R-module isomorphism of R onto PR;n. • Images under ι of left-linear (or right-linear) codes in Rn are left (or right) R-submodules of PR;n. Polynomial Setting Notation Let R be a finite ring with unity. Let x be an indeterminate over R and n a positive integer. • R[x] is the ring of polynomials in x with coefficients in R. • Let hxn − 1i denote the two-sided principal ideal of R[x] n n generated by x − 1. Let PR;n = R[x]=hx − 1i. n n • Define ι : R !PR;n as follows: If c = c0c1 ··· cn−1 2 R , let n−1 n ι(c) = c(x)= c0 + c1x + ··· + cn−1x + hx − 1i. • Images under ι of left-linear (or right-linear) codes in Rn are left (or right) R-submodules of PR;n. Polynomial Setting Notation Let R be a finite ring with unity. Let x be an indeterminate over R and n a positive integer. • R[x] is the ring of polynomials in x with coefficients in R. • Let hxn − 1i denote the two-sided principal ideal of R[x] n n generated by x − 1. Let PR;n = R[x]=hx − 1i. n n • Define ι : R !PR;n as follows: If c = c0c1 ··· cn−1 2 R , let n−1 n ι(c) = c(x)= c0 + c1x + ··· + cn−1x + hx − 1i. Remarks n • Both R and PR;n are left (or right) R-modules under addition and left (or right) scalar multiplication by elements of n R. The map ι is an R-module isomorphism of R onto PR;n. Polynomial Setting Notation Let R be a finite ring with unity. Let x be an indeterminate over R and n a positive integer. • R[x] is the ring of polynomials in x with coefficients in R. • Let hxn − 1i denote the two-sided principal ideal of R[x] n n generated by x − 1. Let PR;n = R[x]=hx − 1i. n n • Define ι : R !PR;n as follows: If c = c0c1 ··· cn−1 2 R , let n−1 n ι(c) = c(x)= c0 + c1x + ··· + cn−1x + hx − 1i. Remarks n • Both R and PR;n are left (or right) R-modules under addition and left (or right) scalar multiplication by elements of n R. The map ι is an R-module isomorphism of R onto PR;n. • Images under ι of left-linear (or right-linear) codes in Rn are left (or right) R-submodules of PR;n. • So images under ι of left-linear (or right-linear) cyclic codes in Rn are left (or right) ideals of PR;n. • We will view left-linear (or right-linear) cyclic codes in either the Rn setting or as left (or right) ideals of PR;n, whichever is convenient. • Polynomials c(x) = c0 + c1x + · · · 2 R[x] will be written without n bold face font; c(x) = c(x) + hx − 1i 2 PR;n. We will say c(x) and c(x) correspond. • Simplification: We write cosets n−1 n c(x) = c0 + c1x + ··· + cn−1x + hx − 1i of PR;n without n hx − 1i; so a(x)b(x) = c(x) 2 PR;n will be written as a polynomial of degree at most n − 1 with the understanding that we really mean n−1 n (a0 + a1x + ··· + an−1x + hx − 1i)(b0 + b1x + ··· + n−1 n n−1 n bn−1x + hx − 1i) = c0 + c1x + ··· + cn−1x + hx − 1i. Polynomial Setting (cont.) Remarks 0 • For c = c0c1 ··· cn−1, let c = cn−1c0 ··· cn−2.