OutlineIntroduction

Definition of nth-Root CodeComputing the Distance

Using a Smaller IdealThe Combinatorial Problem

Future Work

On the Weight Distribution of N-th Root Codes

Fabrizio Caruso Marta Giorgetti

June 10, 2009

Fabrizio Caruso, Marta Giorgetti On the Weight Distribution of N-th Root Codes

OutlineIntroduction

Definition of nth-Root CodeComputing the Distance

Using a Smaller IdealThe Combinatorial Problem

Future Work

Introduction

Definition of nth-Root Code

Computing the Distance

Using a Smaller Ideal

The Combinatorial ProblemThe RecurrenceProof by Generating FunctionsA Computer-Generated Proof

Computing Some Steps of the RecurrenceA Computer-Provable IdentityProving the Guessed Closed Form

Efficiency Considerations

Future Work

Fabrizio Caruso, Marta Giorgetti On the Weight Distribution of N-th Root Codes

OutlineIntroduction

Definition of nth-Root CodeComputing the Distance

Using a Smaller IdealThe Combinatorial Problem

Future Work

A Generalization of Cyclic Codes

I Cyclic codes are a very important class of codes since theyhave a rich algebraic structure, permitting both fast and sharpestimates on their most important parameters.

I In [Gio06], [GS09] the nth-root codes have been introduced.This class is a generalization of the class of cyclic codes and arepresentation of all non-trivial linear codes.

Fabrizio Caruso, Marta Giorgetti On the Weight Distribution of N-th Root Codes

OutlineIntroduction

Definition of nth-Root CodeComputing the Distance

Using a Smaller IdealThe Combinatorial Problem

Future Work

A Generalization of Cyclic Codes

I Cyclic codes are a very important class of codes since theyhave a rich algebraic structure, permitting both fast and sharpestimates on their most important parameters.

I In [Gio06], [GS09] the nth-root codes have been introduced.This class is a generalization of the class of cyclic codes and arepresentation of all non-trivial linear codes.

Fabrizio Caruso, Marta Giorgetti On the Weight Distribution of N-th Root Codes

OutlineIntroduction

Definition of nth-Root CodeComputing the Distance

Using a Smaller IdealThe Combinatorial Problem

Future Work

A Generalization of Cyclic Codes

I Cyclic codes are a very important class of codes since theyhave a rich algebraic structure, permitting both fast and sharpestimates on their most important parameters.

I In [Gio06], [GS09] the nth-root codes have been introduced.This class is a generalization of the class of cyclic codes and arepresentation of all non-trivial linear codes.

Fabrizio Caruso, Marta Giorgetti On the Weight Distribution of N-th Root Codes

OutlineIntroduction

Definition of nth-Root CodeComputing the Distance

Using a Smaller IdealThe Combinatorial Problem

Future Work

Our Approach

I We construct an ideal for each possible weight of thecodewords. We compute the Grobner bases of such ideals andcount the number of corresponding solutions.

I In this paper we only consider binary nth-root codes becauseof the special structure of the ideals describing them.

I We speed up the computation by considering the Grobnerbasis of a smaller ideal and by counting the spurious solutions.

Fabrizio Caruso, Marta Giorgetti On the Weight Distribution of N-th Root Codes

OutlineIntroduction

Definition of nth-Root CodeComputing the Distance

Using a Smaller IdealThe Combinatorial Problem

Future Work

Our Approach

I We construct an ideal for each possible weight of thecodewords. We compute the Grobner bases of such ideals andcount the number of corresponding solutions.

I In this paper we only consider binary nth-root codes becauseof the special structure of the ideals describing them.

I We speed up the computation by considering the Grobnerbasis of a smaller ideal and by counting the spurious solutions.

Fabrizio Caruso, Marta Giorgetti On the Weight Distribution of N-th Root Codes

OutlineIntroduction

Definition of nth-Root CodeComputing the Distance

Using a Smaller IdealThe Combinatorial Problem

Future Work

Our Approach

I We construct an ideal for each possible weight of thecodewords. We compute the Grobner bases of such ideals andcount the number of corresponding solutions.

I In this paper we only consider binary nth-root codes becauseof the special structure of the ideals describing them.

I We speed up the computation by considering the Grobnerbasis of a smaller ideal and by counting the spurious solutions.

Fabrizio Caruso, Marta Giorgetti On the Weight Distribution of N-th Root Codes

OutlineIntroduction

Definition of nth-Root CodeComputing the Distance

Using a Smaller IdealThe Combinatorial Problem

Future Work

Our Approach

I We construct an ideal for each possible weight of thecodewords. We compute the Grobner bases of such ideals andcount the number of corresponding solutions.

I In this paper we only consider binary nth-root codes becauseof the special structure of the ideals describing them.

I We speed up the computation by considering the Grobnerbasis of a smaller ideal and by counting the spurious solutions.

Fabrizio Caruso, Marta Giorgetti On the Weight Distribution of N-th Root Codes

OutlineIntroduction

Definition of nth-Root CodeComputing the Distance

Using a Smaller IdealThe Combinatorial Problem

Future Work

Preliminaries

I We denote by q a power of a prime and n an natural numbersuch (n, q) = 1.

I Let m ∈ N such that the field Fqm is the smallest extensionfield of Fq containing all the zeros of xn − 1. We denote theset of all the n-th roots of unity by Rn.

Fabrizio Caruso, Marta Giorgetti On the Weight Distribution of N-th Root Codes

OutlineIntroduction

Definition of nth-Root CodeComputing the Distance

Using a Smaller IdealThe Combinatorial Problem

Future Work

Preliminaries

I We denote by q a power of a prime and n an natural numbersuch (n, q) = 1.

I Let m ∈ N such that the field Fqm is the smallest extensionfield of Fq containing all the zeros of xn − 1. We denote theset of all the n-th roots of unity by Rn.

Fabrizio Caruso, Marta Giorgetti On the Weight Distribution of N-th Root Codes

OutlineIntroduction

Definition of nth-Root CodeComputing the Distance

Using a Smaller IdealThe Combinatorial Problem

Future Work

Preliminaries

I We denote by q a power of a prime and n an natural numbersuch (n, q) = 1.

I Let m ∈ N such that the field Fqm is the smallest extensionfield of Fq containing all the zeros of xn − 1. We denote theset of all the n-th roots of unity by Rn.

Fabrizio Caruso, Marta Giorgetti On the Weight Distribution of N-th Root Codes

OutlineIntroduction

Definition of nth-Root CodeComputing the Distance

Using a Smaller IdealThe Combinatorial Problem

Future Work

DefinitionLet L ⊂ Rn ∪ 0, L = l1, . . . , lN and P = g1(x), . . . , gr (x) asubset of Fqm [x ] such that ∀ i = 1, . . . ,N there is at least onej = 1, . . . , r such that gj(li ) 6= 0. We denote byC = Ω(q, n, qm, L,P) the code defined over Fq having

H =

g1(l1), . . . , g1(lN)g2(l1), . . . , g2(lN)

......

gr (l1) . . . , gr (lN)

as its parity-check matrix. We say that C is an nth-root code.

Fabrizio Caruso, Marta Giorgetti On the Weight Distribution of N-th Root Codes

OutlineIntroduction

Definition of nth-Root CodeComputing the Distance

Using a Smaller IdealThe Combinatorial Problem

Future Work

ExampleLet q = 2, n = 5, qm = 24, F16 = 〈a〉 ∪ 0, with minimalpolynomial x4 + x + 1 andP = g1(z) = a5z4 + a10z3 + a10z2 + a5z + 1, g2(z) =az4 + a2z3 + a8z2 + a4z + 1, g3(z) = a8z4 + az3 + a4z2 + a2z.The five fifth roots of unity are R5 = a3, a6, a9, a12, 1. Thenth-root code C = Ω(2, 5, 16,R5, g1, g2, g3) is zerofree (0 /∈ L)and its parity-check matrix H is the following: g1(a

3) g1(a6) g1(a

9) g1(a12) g1(1)

g2(a3) g2(a

6) g2(a9) g2(a

12) g2(1)g3(a

3) g3(a6) g3(a

9) g3(a12) g3(1)

=

1 0 0 1 10 1 0 1 10 0 1 1 0

.

Fabrizio Caruso, Marta Giorgetti On the Weight Distribution of N-th Root Codes

OutlineIntroduction

Definition of nth-Root CodeComputing the Distance

Using a Smaller IdealThe Combinatorial Problem

Future Work

DefinitionLet C = Ω(2, n, 2m, L,P) be a binary zerofree nth-root code, w bea natural number such that 2 ≤ w ≤ N. We denote by Jw (C ) thefollowing ideal Jw = Jw (C ) ⊂ F2m [z1, . . . , zw ] , defined by

〈

w∑

h=1

gs(zh)

1≤s≤r

,

znj − 1∏

l∈L(zj − l)

1≤j≤w

, pij(zi , zj)1≤i<j≤w 〉,

where pij =∑n−1

h=0 zhi zn−1−h

j =zni −zn

j

zi−zj∈ F2m [zi , zj ].

RemarkIn the non-binary case the ideal would involve twice as manyvariables and its polynomials would have a less symmetric form.

Fabrizio Caruso, Marta Giorgetti On the Weight Distribution of N-th Root Codes

OutlineIntroduction

Definition of nth-Root CodeComputing the Distance

Using a Smaller IdealThe Combinatorial Problem

Future Work

Since the number of solutions of an ideal J is directly computedfrom any Grobner basis of J (see [BCRT93]) we can obtain easilyan algorithm to compute the weight distribution from the followingproposition.

Proposition

Let C = Ω(2, n, 2m, L,P) be a binary zerofree nth-root code.There is at least one codeword of weight w in C if and only ifthere exists at least one solution of Jw (C ). Moreover the numberof codewords of weight w is Aw = |V(Jw )|/w !.

Fabrizio Caruso, Marta Giorgetti On the Weight Distribution of N-th Root Codes

OutlineIntroduction

Definition of nth-Root CodeComputing the Distance

Using a Smaller IdealThe Combinatorial Problem

Future Work

Using a Smaller Ideal

As proposed in [GS09] one can accelerate the computation ofGrobner basis G of the ideal Jw (C ) by removing the polynomialspi ,j(zi , zj), which guarantee zi 6= zj for any i 6= j . This introducesspurious solutions, that may be counted, with combinatorialarguments, in a recursive way.

DefinitionLet C = Ω(2, n, 2m, L,P) be a binary zerofree nth-root code. Wedenote by Iw = Iw (C ) the following ideal in F2m [Z ]

Iw =⟨ w∑

k=1

gt(zk)

1≤t≤r,

znj − 1∏

l∈L(zj − l)

1≤j≤w

⟩

Fabrizio Caruso, Marta Giorgetti On the Weight Distribution of N-th Root Codes

OutlineIntroduction

Definition of nth-Root CodeComputing the Distance

Using a Smaller IdealThe Combinatorial Problem

Future Work

DefinitionLet V = (z1, . . . , zw ) ∈ (F)w , with w ≥ 2. We say that V isstrongly double-coordinate (sdc) if w is even and for any i ,1 ≤ i ≤ w , |h | zh = zi| is even.

Example

I (α, β, β, γ, β, α, γ, β) is sdc

I (α, α, β, γ, β, γ, γ, α) is not sdc

DefinitionIf w ≥ 2 is even, we say that Jw is a sdc ideal if all its solutionsare sdc.

Fabrizio Caruso, Marta Giorgetti On the Weight Distribution of N-th Root Codes

OutlineIntroduction

Definition of nth-Root CodeComputing the Distance

Using a Smaller IdealThe Combinatorial Problem

Future Work

TheoremFor any 1 ≤ w ≤ d − 1 the ideal Iw (C ) is sdc.Moreover:

I if d is odd, V(Id(C )) = V(Jd(C )),

I if d is even, V(Id(C )) = V(Jd(C )) t Sd , where Sd is the setof all sdc vectors in (V( zn−1Q

l∈L(z−l)))d ∩ (F2m)d

Fabrizio Caruso, Marta Giorgetti On the Weight Distribution of N-th Root Codes

OutlineIntroduction

Definition of nth-Root CodeComputing the Distance

Using a Smaller IdealThe Combinatorial Problem

Future Work

TheoremFor any 1 ≤ w ≤ d − 1 the ideal Iw (C ) is sdc.Moreover:

I if d is odd, V(Id(C )) = V(Jd(C )),

I if d is even, V(Id(C )) = V(Jd(C )) t Sd , where Sd is the setof all sdc vectors in (V( zn−1Q

l∈L(z−l)))d ∩ (F2m)d

Fabrizio Caruso, Marta Giorgetti On the Weight Distribution of N-th Root Codes

OutlineIntroduction

Definition of nth-Root CodeComputing the Distance

Using a Smaller IdealThe Combinatorial Problem

Future Work

TheoremFor any 1 ≤ w ≤ d − 1 the ideal Iw (C ) is sdc.Moreover:

I if d is odd, V(Id(C )) = V(Jd(C )),

I if d is even, V(Id(C )) = V(Jd(C )) t Sd , where Sd is the setof all sdc vectors in (V( zn−1Q

l∈L(z−l)))d ∩ (F2m)d

Fabrizio Caruso, Marta Giorgetti On the Weight Distribution of N-th Root Codes

OutlineIntroduction

Definition of nth-Root CodeComputing the Distance

Using a Smaller IdealThe Combinatorial Problem

Future Work

Corollary

Let C = Ω(2, n, 2m, L,P) be a binary zerofree nth-root code. ThenAd is:

Ad =|V(Id)(C )| − E (d/2,N)

d!(d even),

Ad =|V(Id)(C )|

d!(d odd).

where E (λ, γ) is the number of sdc 2λ-uples with values in1, . . . , γ.

Fabrizio Caruso, Marta Giorgetti On the Weight Distribution of N-th Root Codes

OutlineIntroduction

Definition of nth-Root CodeComputing the Distance

Using a Smaller IdealThe Combinatorial Problem

Future Work

Example

Let C = Ω(2, 255, 28, L,P) be the binary nth-root code such thatL = F256 \ 0 and P = x , x2, x3, x4, x5, x6 ⊂ F2[x ]. C cannothave words of weight 5.By computing a Grobner basis of I5(C ) and J5(C ), we obtain asexpected |I5(C )| = |J5(C )| = 0, but I5(C ) takes less than 3seconds, whereas J5(C ) takes 17 seconds.By computing a Grobner basis of I6(C ) and J6(C ), we obtain that|I6(C )| = 246773955 and |J6(C )| = 0, so that A6(C ) = 0, asexpected, but I6(C ) takes 381.42 seconds, while J6(C ) takes1803.89 seconds.

Fabrizio Caruso, Marta Giorgetti On the Weight Distribution of N-th Root Codes

OutlineIntroduction

Definition of nth-Root CodeComputing the Distance

Using a Smaller IdealThe Combinatorial Problem

Future Work

Proof by Generating FunctionsA Computer-Generated ProofEfficiency Considerations

The Combinatorial Problem

We are interested in computing E (λ, γ) efficiently. We can do thisin different ways: by recurrences or by explicit formulae.

Property

For any positive integers λ and γ we have:

E (λ, γ) =λ∑

k=0

((2λ

2k

)E (k, γ − 1)

).

Moreover we have E (0, γ) = 1, ∀γ ∈ N0; E (λ, 0) = δλ,0, whereδλ,0 is the Kronecker delta.

Fabrizio Caruso, Marta Giorgetti On the Weight Distribution of N-th Root Codes

OutlineIntroduction

Definition of nth-Root CodeComputing the Distance

Using a Smaller IdealThe Combinatorial Problem

Future Work

Proof by Generating FunctionsA Computer-Generated ProofEfficiency Considerations

FactE (λ, γ) can be expressed in terms of integer compositions of λ oflength γ and multinomial coefficients. For λ, γ ≥ 1 one can easilyprove that

E (λ, γ) =∑

(a1,...,aγ)|λ=a1+···+aγ ,

0≤a1,...,aγ≤γ

(2λ

2a1 2a2 . . . 2aγ

)

In particular one can prove that coshγ is the generating functionfor E (λ, γ)/λ!, i.e.

E (λ, γ)/λ! = [xλ] coshγ(x).

from which one can derive a closed form.Fabrizio Caruso, Marta Giorgetti On the Weight Distribution of N-th Root Codes

OutlineIntroduction

Definition of nth-Root CodeComputing the Distance

Using a Smaller IdealThe Combinatorial Problem

Future Work

Proof by Generating FunctionsA Computer-Generated ProofEfficiency Considerations

An alternative strategy for the solution of the combinatorialproblem is

1. unroll the recurrence and compute some steps(fully algorithmic);

2. guess how the recurrence acts on the previous result;

3. formally prove the guessed formula(elementary inductive proof).

Fabrizio Caruso, Marta Giorgetti On the Weight Distribution of N-th Root Codes

OutlineIntroduction

Definition of nth-Root CodeComputing the Distance

Using a Smaller IdealThe Combinatorial Problem

Future Work

Proof by Generating FunctionsA Computer-Generated ProofEfficiency Considerations

An alternative strategy for the solution of the combinatorialproblem is

1. unroll the recurrence and compute some steps(fully algorithmic);

2. guess how the recurrence acts on the previous result;

3. formally prove the guessed formula(elementary inductive proof).

Fabrizio Caruso, Marta Giorgetti On the Weight Distribution of N-th Root Codes

OutlineIntroduction

Definition of nth-Root CodeComputing the Distance

Using a Smaller IdealThe Combinatorial Problem

Future Work

Proof by Generating FunctionsA Computer-Generated ProofEfficiency Considerations

An alternative strategy for the solution of the combinatorialproblem is

1. unroll the recurrence and compute some steps(fully algorithmic);

2. guess how the recurrence acts on the previous result;

3. formally prove the guessed formula(elementary inductive proof).

Fabrizio Caruso, Marta Giorgetti On the Weight Distribution of N-th Root Codes

OutlineIntroduction

Definition of nth-Root CodeComputing the Distance

Using a Smaller IdealThe Combinatorial Problem

Future Work

Proof by Generating FunctionsA Computer-Generated ProofEfficiency Considerations

An alternative strategy for the solution of the combinatorialproblem is

1. unroll the recurrence and compute some steps(fully algorithmic);

2. guess how the recurrence acts on the previous result;

3. formally prove the guessed formula(elementary inductive proof).

Fabrizio Caruso, Marta Giorgetti On the Weight Distribution of N-th Root Codes

OutlineIntroduction

Definition of nth-Root CodeComputing the Distance

Using a Smaller IdealThe Combinatorial Problem

Future Work

Proof by Generating FunctionsA Computer-Generated ProofEfficiency Considerations

In order to compute, say the first 2 steps of the recurrence (i.e. forγ = 1, 2), we could use Maxima as follows

load(zeilbeger);Zeilberger(binomial(2*L,2*k),k,L);

which yields a recurrence which gives 22L−1,

Zeilberger(binomial(2*L,2*k)*2^2*k-1,k,L);

which yields a recurrence whose solution is 1/4(32L − 3).By looking at these two steps we realize that we must find anexpression for the sums of products of the binomial coefficient

(2λ2k

)and 2k-th powers of integers. This leads us to the next step.

Fabrizio Caruso, Marta Giorgetti On the Weight Distribution of N-th Root Codes

OutlineIntroduction

Definition of nth-Root CodeComputing the Distance

Using a Smaller IdealThe Combinatorial Problem

Future Work

Proof by Generating FunctionsA Computer-Generated ProofEfficiency Considerations

Property

For any positive integer λ and α we have:

λ∑k=0

(2λ

2k

)α2k =

(α + 1)2λ + (α− 1)2λ

2.

In order to compute the recurrence and at the same time get acomputer-generated proof of it in English, it is enough to executethe following Maxima commands

load(zeilberger);sm : Zeilberger(binomial(2*L,2*k)*a^(2*k),k,L);zb_prove(binomial(2*L,2*k)*a^(2*k),k,L,sm);

Fabrizio Caruso, Marta Giorgetti On the Weight Distribution of N-th Root Codes

OutlineIntroduction

Definition of nth-Root CodeComputing the Distance

Using a Smaller IdealThe Combinatorial Problem

Future Work

Proof by Generating FunctionsA Computer-Generated ProofEfficiency Considerations

We can now prove and find a general form for E (λ, γ):

TheoremFor any positive integer λ and γ we have:

E (λ, γ) = 2−γγ∑

i=0

(γ

i

)(γ − 2i)2λ = 2−γ+1

dγ/2e−1∑i=0

(γ

i

)(γ − 2i)2λ

Proof.We prove the theorem by induction on γ.

Fabrizio Caruso, Marta Giorgetti On the Weight Distribution of N-th Root Codes

OutlineIntroduction

Definition of nth-Root CodeComputing the Distance

Using a Smaller IdealThe Combinatorial Problem

Future Work

Proof by Generating FunctionsA Computer-Generated ProofEfficiency Considerations

Complexity

The recurrence allows us to compute in reasonable time just a fewvalues of E (λ, γ) for very small values of λ and γ.

I The number of arithmetic operations if the recurrence is usedis a function in Θ(

(λ+γ

γ

)).

I The total number of arithmetic operations needed to computeE (λ, γ) through the explicit formula given is O(γ log(λ)).

Fabrizio Caruso, Marta Giorgetti On the Weight Distribution of N-th Root Codes

OutlineIntroduction

Definition of nth-Root CodeComputing the Distance

Using a Smaller IdealThe Combinatorial Problem

Future Work

Proof by Generating FunctionsA Computer-Generated ProofEfficiency Considerations

Complexity

The recurrence allows us to compute in reasonable time just a fewvalues of E (λ, γ) for very small values of λ and γ.

I The number of arithmetic operations if the recurrence is usedis a function in Θ(

(λ+γ

γ

)).

I The total number of arithmetic operations needed to computeE (λ, γ) through the explicit formula given is O(γ log(λ)).

Fabrizio Caruso, Marta Giorgetti On the Weight Distribution of N-th Root Codes

OutlineIntroduction

Definition of nth-Root CodeComputing the Distance

Using a Smaller IdealThe Combinatorial Problem

Future Work

Proof by Generating FunctionsA Computer-Generated ProofEfficiency Considerations

Complexity

The recurrence allows us to compute in reasonable time just a fewvalues of E (λ, γ) for very small values of λ and γ.

I The number of arithmetic operations if the recurrence is usedis a function in Θ(

(λ+γ

γ

)).

I The total number of arithmetic operations needed to computeE (λ, γ) through the explicit formula given is O(γ log(λ)).

Fabrizio Caruso, Marta Giorgetti On the Weight Distribution of N-th Root Codes

OutlineIntroduction

Definition of nth-Root CodeComputing the Distance

Using a Smaller IdealThe Combinatorial Problem

Future Work

Future Work

Some of the things that remain to be done:

I Extending this method to the whole distribution of weights(done but needs to be checked)

I Applying this approach to Hermitian Codes and other codes(probably easy but it might not pay off as much)

Fabrizio Caruso, Marta Giorgetti On the Weight Distribution of N-th Root Codes

OutlineIntroduction

Definition of nth-Root CodeComputing the Distance

Using a Smaller IdealThe Combinatorial Problem

Future Work

Future Work

Some of the things that remain to be done:

I Extending this method to the whole distribution of weights(done but needs to be checked)

I Applying this approach to Hermitian Codes and other codes(probably easy but it might not pay off as much)

Fabrizio Caruso, Marta Giorgetti On the Weight Distribution of N-th Root Codes

OutlineIntroduction

Definition of nth-Root CodeComputing the Distance

Using a Smaller IdealThe Combinatorial Problem

Future Work

Future Work

Some of the things that remain to be done:

I Extending this method to the whole distribution of weights(done but needs to be checked)

I Applying this approach to Hermitian Codes and other codes(probably easy but it might not pay off as much)

Fabrizio Caruso, Marta Giorgetti On the Weight Distribution of N-th Root Codes

OutlineIntroduction

Definition of nth-Root CodeComputing the Distance

Using a Smaller IdealThe Combinatorial Problem

Future Work

[BCRT93] A. M. Bigatti, P. Conti, L. Robbiano, and C. Traverso.A “divide and conquer” algorithm for Hilbert-Poincareseries, multiplicity and dimension of monomial ideals.In Applied algebra, algebraic algorithms anderror-correcting codes (San Juan, PR, 1993), volume673 of Lecture Notes in Comput. Sci., pages 76–88.Springer, Berlin, 1993.

[Gio06] M. Giorgetti.On some algebraic interpretation of classical codes.PhD thesis, University of Milan, 2006.

[GS09] M. Giorgetti and M. Sala.A commutative algebra approach to linear codes.Journal of Algebra, 321(8):2259–2286, 2009.

Fabrizio Caruso, Marta Giorgetti On the Weight Distribution of N-th Root Codes