Affine-invariant codes

In this section we will look at extending certain cyclic codes and examine an important class of codes called affine-invariant codes. Reed–Muller codes and some BCH codes, defined in Chapter 5, are affine-invariant.

We will present a new setting for “primitive” cyclic codes that will assist us in the description of affine-invariant codes. Aprimitive cyclic code overFq is a cyclic code of lengthn =q^t−1 for somet.¹To proceed, we need some notation.

LetIdenote the field of orderq^t, which is then an extension field ofFq. The setIwill be the index set of our extended cyclic codes of lengthq^t. LetI^∗be the nonzero elements ofI, and supposeαis a primitiventh root of unity inI(and hence a primitive element of I). The setI^∗will be the index set of our primitive cyclic codes of lengthn=q^t−1. With

The setFq[I] is actually an algebra under the operations

c

1 Cyclic codes of lengthn=q^t−1 over an extension field ofFqare also called primitive, but we will only study the more restricted case.

163 4.7 Affine-invariant codes

The vector spaceFq[I^∗] will be the new setting for primitive cyclic codes, and the algebra Fq[I] will be the setting for the extended cyclic codes. So in fact both codes are contained inFq[I], which makes the discussion of affine-invariant codes more tractable.

Suppose thatCis a cyclic code overFqof lengthn=q^t−1. The coordinates ofChave been denoted{0,1, . . . ,n−1}. InRn, theith componentciof a codewordc=c0c1· · ·cn−1, with associated polynomialc(x), is the coefficient of the termcixⁱinc(x); the component ci is kept in positionxⁱ. Now we associatecwith an elementC(X)∈Fq[I^∗] as follows: We now need to examine the cyclic shiftxc(x) under the correspondence (4.9). We have xc(x)=cn−1+ Example 4.7.2 We continue with Example 4.7.1. Namely, xc(x)=x+x²+x⁴↔X^α+X^α2+X^α4=X^α1+X^αα+X^αα3.

The coordinates of our cyclic codes are indexed byI^∗. We are now ready to extend our cyclic codes. We use the element 0∈I to index the extended coordinate. The extended codeword ofC(X)= From (4.10), together with the observation that X^α0 =X⁰=1, in this terminology an extended cyclic code is a subspaceCofFq[I] such that

With this new notation we want to see where the concepts of zeros and defining sets come in. This can be done with the assistance of a functionφs. LetN = {s|0≤s≤n}. codeC defined inRn had defining setT relative to thenth root of unity α. Then (4.11) shows that if 1≤s≤n−1,s∈T if and only ifφs(C(X))=0 for allC(X)∈C. Finally, what isφn(C(X))? Equation (4.11) works in this case as well, implying thatαⁿ =1 is a zero ofCif and only ifφn(C(X))=0 for allC(X)∈C. Butα⁰=αⁿ=1. Hence we have, by Exercise 238, that 0∈T if and only ifφn(C(X))=0 for allC(X)∈C. We can now describe an extended cyclic code in terms of a defining set as follows: a codeCof lengthq^t is anextended cyclic code with defining setT providedT ⊆Nis a union ofq-cyclotomic cosets modulon=q^t−1 with 0∈T and

C= {C(X)∈Fq[I]|φs(C(X))=0 for alls∈T}. (4.12) We make several remarks.

r 0 andnare distinct inT and each is its ownq-cyclotomic coset.

r 0∈T because we need all codewords inCto be even-like.

r Ifn∈T, thenφn(C(X))=

g∈I^∗Cg=0. As the extended codeword is even-like, since 0∈T,

g∈ICg =0. Thus the extended coordinate is always 0, a condition that makes the extension trivial.

r The defining setTofCand the defining setT are closely related:Tis obtained by taking T, changing 0 tonif 0∈T, and adding 0.

165 4.7 Affine-invariant codes

r s∈T with 1≤s≤nif and only ifα^sis a zero ofC.

r By employing the natural ordering 0, αⁿ, α¹, . . . , αⁿ⁻¹onI, we can make the extended cyclic nature of these codes apparent. The coordinate labeled 0 is the parity check co-ordinate, and puncturing the codeCon this coordinate gives the codeCthat admits the standard cyclic shiftαⁱ →ααⁱ as an automorphism.

r To check that a codeCinRnis cyclic with defining setT, one only needs to verify that of F8 satisfying α³=1+α. See Examples 3.4.3 and 4.3.4. The extended generator is 1+X+X^α+X^α3. Then a generator matrix forCis In Section 1.6, we describe how permutation automorphisms act on codes. From the discussion in that section, a permutationσ ofIacts onCas follows:

g∈I ag+b. Notice that the maps σa,0 are merely the cyclic shifts on the coordinates {αⁿ, α¹, . . . , αⁿ⁻¹}each fixing the coordinate 0. The group GA1(I) has order (n+1)n = q^t(q^t−1). An affine-invariant code is an extended cyclic codeC over Fq such that GA1(I)⊆PAut(C). Amazingly, we can easily decide which extended cyclic codes are affine-invariant by examining their defining sets. In order to do this we introduce a partial ordering' onN. Suppose thatq =p^m, where p is a prime. ThenN = {0,1, . . . ,n}, expansion ofr. Notice that ifr 's, then in particularr≤s. We also need a result called Lucas’ Theorem [209], a proof of which can be found in [18].

Theorem 4.7.5 (Lucas) Let r =mt−1

We are now ready to determine the affine-invariant codes from their defining sets, a result due to Kasami, Lin, and Peterson [162].

Theorem 4.7.6 LetCbe an extended cyclic code of length q^twith defining setT . The code Cis affine-invariant if and only if whenever s∈T then r ∈T for all r ∈N with r 's.

Conversely, assume that C is affine-invariant. Assume that s∈T and that r's.

We need to show that r∈T, that is φr(C(X))=0 by (4.12). AsC is affine-invariant,

for allb∈I. But the right-hand side of this equation is a polynomial inb of degree at mosts<q^twith allq^tpossibleb∈Ias roots. Hence this must be the zero polynomial. So (^s_r)φr(C(X))=0 inIfor allr's. However, by Lucas’ Theorem again, (^s_r)≡0 (mod p) and thus these binomial coefficients are nonzero inI. Henceφr(C(X))=0 implying that

r∈T.

Example 4.7.7 Suppose thatC is an affine-invariant code with defining setT such that n∈T. By Exercise 279,r'nfor allr∈N. ThusT =N andC is the zero code. This makes sense because ifn ∈T, then the extended component of any codeword is always

167 4.7 Affine-invariant codes

0. Since the code is affine-invariant and the group GA₁(I) is transitive (actually doubly-transitive), every component of any codeword must be zero. ThusCis the zero code.

Exercise 279 Prove thatr 'nfor allr∈N.

Example 4.7.8 The following table gives the defining sets for the binary extended cyclic codes of length 8. The ones that are affine-invariant are marked.

Defining set Affine-invariant

{0} yes

{0,1,2,4} yes {0,1,2,3,4,5,6} yes {0,1,2,4,7} no {0,1,2,3,4,5,6,7} yes

{0,3,5,6} no {0,3,5,6,7} no

{0,7} no

Notice the rather interesting phenomenon that the extended cyclic codes with defining sets{0,1,2,4}and{0,3,5,6}are equivalent, yet one is affine-invariant while the other is not. The equivalence of these codes follows from Exercise 41 together with the fact that the punctured codes of length 7 are cyclic with defining sets{1,2,4}and{3,5,6}and are equivalent under the multiplierµ3. These two extended codes have isomorphic auto-morphism groups. The autoauto-morphism group of the affine-invariant one contains GA1(F8) while the other contains a subgroup isomorphic, but not equal, to GA₁(F8). The code with defining set{0}is the [8,7,2] code consisting of the even weight vectors inF⁸₂. The code with defining set{0,1,2,3,4,5,6}is the repetition code, and the code with defining set {0,1,2,3,4,5,6,7}is the zero code. Finally, the code with defining set{0,1,2,4}is one

particular form of the extended Hamming codeH3.

Exercise 280 Verify all the claims in Example 4.7.8.

Exercise 281 Give the defining sets of all binary affine-invariant codes of length 16.

Exercise 282 Give the defining sets of all affine-invariant codes overF4of length 16.

Exercise 283 Give the defining sets of all affine-invariant codes overF3of length 9.

Corollary 4.7.9 IfCis a primitive cyclic code such thatCis a nonzero affine-invariant code, then the minimum weight ofCis its minimum odd-like weight. In particular,the minimum weight of a binary primitive cyclic code,whose extension is affine-invariant,is odd.

Proof: As GA1(I) is transitive, the result follows from Theorem 1.6.6.

In this chapter we examine one of the many important families of cyclic codes known as BCH codes together with a subfamily of these codes called Reed–Solomon codes.

The binary BCH codes were discovered around 1960 by Hocquenghem [132] and in-dependently by Bose and Ray-Chaudhuri [28, 29], and were generalized to all finite fields by Gorenstein and Zierler [109]. At about the same time as BCH codes appeared in the literature, Reed and Solomon [294] published their work on the codes that now bear their names. These codes, which can be described as special BCH codes, were actually first constructed by Bush [42] in 1952 in the context of orthogonal arrays. Because of their burst error-correction capabilities, Reed–Solomon codes are used to improve the reliability of compact discs, digital audio tapes, and other data storage systems.