Closed formulas for the error locator polynomials of cyclic codes - PDF document

Closed formulas for the error locator polynomials of cyclic codes Dedicated to Tom Høholdt in honour of his 60-th birthday by Ruud Pellikaan Discrete Mathematics Technical University of Eindhoven June 2, 2005

Decoding cyclic codes beyond the BCH error-correcting capacity. Roughly four methods: I) Buchberger’s algorithm, Gr¨ obner basis syndrome equations in error-positions and values II) Newton identities in syndromes and coefficients of error-locator polynomial III) Majority voting of unknown syndromes, error-locating pairs IV) Sudan’s list decoding

Ternary Golay code [11,6,5] Cyclic code of length n = 11 over F 3 11 divides 3 5 − 1: 11 ∗ 22 = 242 α ∈ F 243 of order 11 c ( x ) = c 0 + c 1 x + · · · + c 10 x 10 with c i ∈ F 3 and x 11 = 1 is a codeword if and only if c ( α ) = 0

if c ( x )is a codeword, then c ( α ) = 0 so c ( α 3 ) = 0, c ( α 9 ) = 0, . . . Hence c ( α i ) = 0 for i one of 1 , 3 , 9 , 27 ≡ 5 , 15 ≡ 4 , 12 ≡ 1 that is 1 , − , 3 , 4 , 5 , − , − , − , 9 , − , −

complete defining set 1 , − , 3 , 4 , 5 , − , − , − , 9 , − , − hence the dimension is 11 − 5 = 6 minimum distance d is at least d BCH = 4 in fact d = 5 so 2 error-correcting

Error-correction c ( x ) the codeword sent over a channel y ( x ) the received word y ( x ) = c ( x ) + e ( x ) with e ( x ) the error vector/polynomial s j = e ( α j ) the j -th syndrome If j in the complete defining set, then s j = y ( α j ) is a known syndrome, otherwise an unknown syndrome

Now s 3 j = s 3 j Hence s 3 s 9 1 = s 3 1 = s 9 s 27 s 81 = s 5 = s 4 1 1 s 243 = s 1 1

in case of t errors error polynomial e ( x ) has weight t supported at positions i 1 < · · · < i t e ( x ) = e i 1 x i 1 + · · · + e i t x i t

e ( α j ) = e i 1 α ji 1 + · · · + e i t α ji t s j = e ( α j ) = y 1 x j 1 + · · · + y t x j t x w = α i w error position y w = e i w error value

Minimum distance d = 5 Assume that number of errors t ≤ 2 Then c ( x ) is the unique closest codeword and s 1 = y 1 x 1 + y 2 x 2 y 3 i = y i and x 11 = 1 for i = 1 , 2 i

Try to solve these nonlinear equations by clever manipulations, or apply Buchberger’s algorithm for a Gr¨ obner basis 1990-1994: Brinton Cooper III and Chen-Helleseth-Reed-Truong

Polynomial equations S 1 = Y 1 X 1 + Y 2 X 2 Y 2 1 = 1 , X 11 = 1 , Y 2 = 1 , X 11 = 1 1 2 ( X 1 − X 2 ) 242 = 1 in the variables with lex order S 1 < X 1 < X 2 < Y 1 < Y 2 elimination order

Computer algebra packages Maple, Mathematica: slow Singular, Magma: fast computing Gr¨ obner bases

Example with Magma P<Y2,Y1,X2,X1,S1> := PolynomialRing(GF(3),5); I := ideal<P|X1*Y1+X2*Y2-S1, Y1^2-1,Y2^2-1,X1^11-1,X2^11-1, (X1-X2)^242-1>; GroebnerBasis(I);

Magma V2.11-10 Fri May 13 2005 08:29:55 on modul ------------------------------------- [ . . . . X1^2 + 2*X1*S1^144 + X1*S1^100 + 2*X1*S1^34 + X1*S1^12 + 2*S1^200 + S1^178 + 2*S1^156 + 2*S1^134 + S1^90 + 2*S1^68 + 2*S1^46 + S1^24 + 2*S1^2, S1^220 + S1^198 + S1^176 + S1^154 + S1^132 + S1^110 + S1^88 + S1^66 + S1^44 + S1^22 + 1 ] Total time: 0.230 seconds, Total memory usage: 3.99MB

X1^2 + 2*X1*S1^144 + X1*S1^100 + 2*X1*S1^34 + X1*S1^12 + 2*S1^200 + S1^178 + 2*S1^156 + 2*S1^134 + S1^90 + 2*S1^68 + 2*S1^46 + S1^24 + 2*S1^2 Coefficient of X 2 1 : 1 Coefficient of X 1 : 2 S 144 + S 100 + 2 S 34 + S 12 1 1 1 1 Coefficient of 1: 2 S 200 + S 178 + 2 S 156 + 2 S 134 + 1 1 1 1 S 90 + 2 S 68 + 2 S 46 + S 24 + 2 S 2 1 1 1 1 1

Closed formula of error-locator polynomial σ ( X ) = ( X − x 1 )( X − x 2 ) = 1 + σ 1 X + σ 2 X 2 − S 144 + S 100 − S 34 + S 12 σ 1 ( S 1 ) = 1 1 1 1 − S 200 + S 178 − S 156 − S 134 σ 2 ( S 1 ) = + 1 1 1 1 S 90 − S 68 − S 46 + S 24 − S 2 1 1 1 1 1

Generic case of cyclic code with 1 , 2 , . . . , 2 t in defining set and received word with at most t errors Syndrome equations : S j = Y 1 X j 1 + Y 2 X j 2 + · · · + Y t X j j = 1 , . . . , 2 t, t Error-locator polynomial t t σ ( t ) σ ( t ) ( X ) = X t − j � � ( X − X i ) = j i =1 j =0

Elementary symmetric functions: σ ( t ) = 1 0 σ ( t ) = − ( X 1 + X 2 + · · · + X t ) 1 . . . σ ( t ) = ( − 1) t X 1 X 2 · · · X t t σ ( t ) = ( − 1) j � X i 1 X i 2 · · · X i t j 1 ≤ i 1 < ··· <i j ≤ t

Peterson-Arimoto use Generalized Newton identities :  S t +1 + σ 1 S t + + σ t S 1 = 0 · · ·    S t +2 + σ 1 S t +1 + + σ t S 2 = 0 · · ·  . . . . . . . . .    S 2 t + σ 1 S 2 t − 1 + + σ t S t = 0 . · · ·  to solve linear equations in the variables σ j with coefficients in the S r .

Matrix equation:   1     S t +1 S t S 1 0 · · · σ 1   S t +2 S t +1 S 2 0   · · ·       σ 2 =  .  . . . .   .  . . . . .   . . . . .     .  .  .      S 2 t S 2 t − 1 S t 0 · · ·   σ t or equivalently  S t S 1   σ 1   S t +1  · · · S t +1 S 2 σ 2 S t +2 · · ·        = −  .  . . .   .   .  . . . . . . . . . .           S 2 t − 1 S t σ t S 2 t · · ·

Case t = 2 � S 2 σ 1 + S 1 σ 2 = − S 3 S 3 σ 1 + S 2 σ 2 = − S 4 Cramer’s rule : − S 3 S 1 − S 4 S 2 = S 1 S 4 − S 2 S 3 σ 1 = S 2 S 2 S 1 2 − S 1 S 3 S 3 S 2 and similarly σ 2 = S 2 3 − S 2 S 4 S 2 2 − S 1 S 3

Generic error-locator polynomial for t = 2 2 − S 1 S 3 ) X 2 + ( S 1 S 4 − S 2 S 3 ) X + ( S 2 ( S 2 3 − S 2 S 4 ) X 2 � � X 1 � � � � = S 3 S 2 S 1 � � � � � S 4 S 3 S 2 � � �

Closed formula of generic error-locator polynomial � � X t X t − 1 1 · · · � � � � S t +1 S t S 1 � · · · � � � σ ( t ) ( X ) = � � S t +2 S t +1 S 2 · · · � � . . . . � � . . . . . . . . � � � � S 2 t S 2 t − 1 S t � · · · � � �

Binary cyclic codes error values are always one: Y i = 1, syndromes: S 2 j = S 2 j equations: S 2 j − 1 = X 2 j − 1 + X 2 j − 1 + · · · + X 2 j − 1 1 2 t for j = 1 , . . . , t

Algebraic system of t equations in 2 t variables. Eliminate the X 2 , . . . , X t . Gives one equation β ( t ) ( X 1 ) = 0 in the variable X 1 with coefficients β ( t ) ( S 1 , S 3 . . . , S 2 t − 1 ) i

For t = 1 β (1) ( X ) = X + S 1 Compare: σ (1) ( X ) = S 1 X − S 2 Reduce mod 2 and S 2 j = S 2 j σ (1) ( X ) = S 1 X + S 2 1 = S 1 β (1) ( X )

For t = 2, β (2) ( X ) = S 1 X 2 + S 2 1 X + ( S 3 + S 3 1 ) Compare: σ (2) ( X ) = 2 ) X 2 − ( S 4 S 1 − S 2 S 3 ) X + ( S 4 S 2 − S 2 ( S 3 S 1 − S 2 3 )

Reduction mod 2 and S 2 j = S 2 j gives 1 ) X 2 + ( S 3 S 3 ( S 3 S 1 + S 4 1 + S 5 1 ) X + ( S 2 3 + S 6 1 ) Divide by S 3 + S 3 1 S 1 X 2 + S 2 1 X + ( S 3 + S 3 1 ) is again β (2) ( X )

For t = 3, β (3) ( X ) = 1 ) X 3 + S 1 ( S 3 + S 3 ( S 3 + S 3 1 ) X 2 + ( S 5 + S 3 S 2 1 ) X + ( S 5 S 1 + S 2 3 + S 3 S 3 1 + S 6 1 ) Compare: σ (3) ( X ) = ( S 5 S 3 S 1 − S 5 S 2 2 − S 2 4 S 1 + 2 S 4 S 3 S 2 − S 3 3 ) X 3 − ( S 6 S 3 S 1 − S 6 S 2 2 − S 5 S 4 S 1 + S 5 S 3 S 2 − S 4 S 2 3 + S 2 4 S 2 ) X 2 +( S 6 S 4 S 1 − S 6 S 3 S 2 − S 2 5 S 1 + S 5 S 2 3 + S 5 S 4 S 2 − S 2 4 S 3 ) X − ( S 6 S 4 S 2 − S 2 3 S 6 − S 2 5 S 2 + 2 S 5 S 4 S 3 − S 3 4 )

Reduction mod 2 and S 2 j = S 2 j gives ( S 5 S 3 S 1 + S 5 S 4 1 + S 3 3 + S 9 1 ) X 3 + ( S 3 3 S 1 + S 5 S 5 1 + S 5 S 3 S 2 1 + S 10 1 ) X 2 + ( S 2 5 S 1 + S 5 S 2 3 + S 5 S 6 1 + S 3 3 S 2 1 + S 2 3 S 5 1 + S 3 S 8 1 ) X + ( S 2 5 S 2 1 + S 4 3 + S 2 3 S 6 1 + S 12 1 ) Division by ( S 3 S 3 1 + S 2 3 + S 5 S 1 + S 6 1 ) gives 1 ) X 3 + S 1 ( S 3 + S 3 ( S 3 + S 3 1 ) X 2 + ( S 5 + S 3 S 2 1 ) X + ( S 5 S 1 + S 2 3 + S 3 S 3 1 + S 6 1 ) is again β (3) ( X )

General result – β ( t ) ( X ) weighted homogeneous of degree t ( t + 1) / 2 – β ( t ) = β ( t +1) t 0 – β ( t ) = S 1 β ( t ) 1 0 – The reduction of σ ( t ) ( X ) modulo 2 and S 2 j = S 2 j is equal to β ( t ) 0 β ( t ) ( X )

Newton identities  S 1 + σ 1 +0 +0 + · · · +0 = 0    S 2 + σ 1 S 1 +2 σ 2 +0 + · · · +0 = 0     S 3 + σ 1 S 2 + σ 2 S 1 +3 σ 3 + · · · +0 = 0 . . . . . . . . .      S t + σ 1 S t − 1 + σ 2 S t − 2 + σ 3 S t − 3 + · · · + tσ t = 0 .    + + + = 0 S t +1 σ 1 S t σ t S 1 · · ·   + + + = 0  S t +2 σ 1 S t +1 σ t S 2 · · ·  . . . . . . . . .    S 2 t + σ 1 S 2 t − 1 + + σ t S t = 0 . · · · 