On the second weight of generalized Reed-Muller codes

On the second weight of generalized Reed-Muller codes

O. Geil

Aalborg University

ACA - July 2008

Error correcting codes - Linear codes

Purpose: Reliable communication over noisy channels.

Definition:

A linear code C is a k -dimensional subspace of Fⁿ_q. Choose an isomorphism from F^k_q to C.

Encode messagem~ = (m₁, . . . ,m_k) ∈F^k_qto its image

~c= (c₁, . . . ,c_n) ∈C (called a codeword).

Error correction

~c is send over channel, but noise~e is added.

Receiver gets:~r = ~c+ ~e.

Idea:

If only few e_i’s are non-zero and if codewords in C are “far” from each other, then receiver can recover~c by choosing codeword

“nearest” to~r .

Weights and distances

~a, ~b∈Fⁿ_q.

w_H(~a) = #{i |a_i 6=0}.

dist_H(~a, ~b) = #{i|a_i 6=b_i} =w_H(~a− ~b).

d = min{w_H(~c) | ~c ∈C\{~0}}

= min{dist_H(~a, ~b) | ~a, ~b∈C, ~a6= ~b}

is called minimum distance (or minimum weight).

If w_H(~e) ≤ b^d−1₂ cthen nearest codeword to~r = ~c+ ~e is~c.

Decoding strategies

I Minimum distance decoding. Radius isb(d−1)/2c.

I List decoding. Radius is larger thanb(d−1)/2c.

I Maximum likelihood decoding. Look for nearest codeword.

Weight distribution is{W₁, . . . ,Wn}where W_i = #{~c ∈C|w_H(~c) =i}.

Weight distribution = Distance distribution.

Weight distribution tells you how good is List decoding and Maximum likelihood decoding.

Generalized Reed-Muller codes

F^m_q = {P₁, . . . ,P_q^m}.

RM_q(s,m) = {(F(P₁), . . . ,F(P_q^m)) |F ∈F_q[X₁, . . . ,X_m], deg(F) ≤s,deg_X

i(F) <q, for i =1, . . . ,m}

Question 1a:

How many zeros can a multivariate q-ary polynomial of total degree s possibly have?

Question 1b:

What is the minimum distance of RM_q(s,m)?

Question 2a:

What is the second highest number of attainable zeros for a multivariate q-ary polynomial of total degree s?

Question 2b:

What is the second weight of RMq(s,m)?

Parameters for RM

(s, m)

I Minimum distance. Kasami, Lin, Peterson 1968. The BCH-bound.

I Various results on weight distribution for q =2.

I For s≤2 weight distribution. McEliece 1969.

I Second weight for small dimensions. Cherdieu, Rolland 1996. Geometric arguments.

I Second weight for s<q/2. Sboui 2007. Geometric arguments.

I Some results on third weight. Rodier, Sboui in 2008.

Geometric arguments.

I Second weight for s<q (and for high values of s). O.G. in 2008. Ideal-variety approach.

I Second weight for all s (except s≡1 mod (q−1)).

Rolland, recent preprint. Ideal-variety approach.

The ideal-variety approach

Find (or estimate)

#V_Fq(hF(X₁, . . . ,Xm)i)

= #V_Fq(hF(X₁, . . . ,X_m),X₁^q−X₁, . . . ,X_m^q−X_mi)

= #VF¯q(hF(X₁, . . . ,X_m),X₁^q−X₁, . . . ,X_m^q−X_mi).

General set-up

Problem: Given I ⊆F_q[X₁, . . . ,X_m]find#V_Fq(I).

Solution:

∆≺(J) = {X₁ⁱ¹· · ·X_m^im |X₁ⁱ¹· · ·X_m^im is not leading monomial of any polynomial in J}

I_q =I+ hX₁^q−X₁, . . . ,X_m^q−X_mi.

#V_Fq(I) = #∆≺(Iq).

An easy proof of #V

(I

) ≤ #∆

(I

)

Given Gröbner basisGthen division with remainder modGis unique. Hence,

{M+J |M ∈ ∆_≺(J)}

is a basis for F_q[X₁, . . . ,X_m]/J.

Let V_Fq(Iq) = {P₁, . . . ,Pn}. The map

ev:Fq[X₁, . . . ,Xm]/Iq→Fⁿ_q ev(F+I_q) = (F(P₁), . . . ,F(P_n)) is a surjective homomorphism.

QED

To see surjective...

Write P_j = (P_j⁽¹⁾, . . . ,P_j^(m)). Then

ev Y

s=1,...,m

Y j=1, . . . ,n P_j^(s)6=P_i^(s)

(X_s−P_j^(s))
+I_q

is nonzero in position i, but zero elsewhere.

Proof of #V

(I

) = #∆

(I

)

Proof of injectivity involves

I radical ideals

I algebraic closure

I strong Nullstellensatz.

Monomial ordering

Mostly for simplicity, assume≺is degree lexicographic ordering.

X₁^a1· · ·X_m^am ≺X₁^b1· · ·X_m^bm if either 1. or 2.

1. a₁+ · · · +a_m<b₁+ · · · +b_m 2. a₁+ · · · +a_m=b₁+ · · · +b_m but

X₁^a1· · ·X_m^am ≺_lex X₁^b1· · ·X_m^bm

Minimum distance of RM

(s, m)

F(X₁, . . . ,X_m) ∈F_q[X₁, . . . ,X_m] deg_X

i(F) <q, i=1, . . . ,m.

lm(F) =X₁ⁱ¹· · ·X_m^im, i₁+ · · · +i_m ≤s.

#∆(hF(X₁, . . . ,Xm),X₁^q−X₁, . . . ,X_m^q −Xmi)

≤ #∆(hX₁ⁱ¹· · ·X_m^im,X₁^q, . . . ,X_m^qi)

= q^m−

m

Y

l=1

(q−i_l)

Minimum distance - cont

Let s=a(q−1) +b, 0≤b<q−1.

i₁+ · · · +i_m ≤s, i₁<q, . . . ,i_m<q max{q^m−Qm

l=1(q−i_l)} =q^m− (q−b)q^m−a−1.

On the other hand, polynomial

(X₁^q−1−1) · · · (X_a^q−1−1)(X_a+1− α₁) · · · (X_a+1− α_b) has this amount of zeros.

Second weight for s < q

Proposition:

If F(X₁, . . . ,X_m)is of total degree s then either sq^m−1zeros or at most sq^m−1− (s−1)q^m−2zeros. Both values can be realized.

lm(F) =X₁ⁱ¹· · ·x_m^im

Case 1: i₁<s, . . . ,i_m <s

#∆(hX₁ⁱ¹· · ·X_m^im,X₁^q, . . . ,X_m^qi)

= q^m−

m

Y

l=1

(q−i_l)

is at most sq^m−1− (s−1)q^m−2

Second weight for s < q, cont.

Case 2: i₁=s, i₂= · · · =im=0

H(X₁, . . . ,X_m) = S(X₁^q−X₁,F(X₁, . . . ,X_m))

= X₁^q−X₁−X₁^q−sF(X₁, . . . ,X_m) Total degree of H at most q.

R(X₁, . . . ,Xm) =

H(X₁, . . . ,X_m)rem (F(X₁, . . . ,X_m),X₁^q−X₁, . . . ,X_m^q −X_m).

If R(X₁, . . . ,Xm) =0 then{F,X₁^q−X₁, . . . ,X_m^q−Xm}a GB and sq^m−1zeros.

case 2 - cont

lm(R) =X₁^v1· · ·X_m^vm. Pm

i=1v_i ≤q, v₁<s, v₂<q, . . . ,v_m<q.

#∆(hF(X₁, . . . ,X_m),X₁^q−X₁, . . . ,X_m^q −X_mi)

≤ #∆(hX₁^s,X₂^q, . . . ,X_m^q,X₁^v1· · ·X_m^vmi)

= sq^m−1− (s−v₁)

m

Y

i=2

(q−v_i)

≤ sq^m−1− (s−1)q^m−2

There are simple polynomials with exactly sq^m−1− (s−1)q^m−2 zeros.

(m − 1)(q − 1) < s

d(RM_q(s−1,m)) =d(RM_q(s,m)) +1 Hence, second weight is d+1

Second weight for RMq(s,2)covered.

...very recently

... Robert Rolland used method from this talk plus a technical lemma to find remaining second weights

...except for the case s=a(q−1) +1...