DUALITY OF CHAINS OF ALMOST AFFINE CODES
Abstract (English):
Almost affine codes are generalization for widely used linear codes which can be used in ideal perfect secret sharing schemes. In [1] Simonis and Ashikhmin defined and studied some properties of almost affine codes. In the other hand quasi-uniform codes [2] are generalization of almost affine codes. In this paper we show that duality of chains of linear codes holds in the almost affine case as well and we make a conjecture about such property for quasi-uniformcodes.

Keywords:
almost affine codes, quasi-uniform codes, matroids

Introduction

Linear block codes play a key role in the theory of error correction. A q -ary linear code of length n  is a k -dimensional vector subspace of  Fqn  where F  is a finite field with q  elements, also known as the alphabet of the code. One of the reasons why linear codes dominate the industry is that linear code can be described completely with its generator matrix. Because of the convenience, linear codes possess some restrictions, and it can be shown [9] that linear codes are not sufficient to achieve the maximal capacity for information flow in a multi-source network.

In [1] Simonis and Ashikhmin proposed another class of error correcting codes, namely almost affine codes. The initial motivation for authors was studying the ideal perfect secret sharing schemas. Basically, almost affine codes are generalization for linear codes with less restrictions. However, its turs out that almost affine codes share some properties with linear codes, like the subject of this paper – duality of chains of codes. Or specifically the demi-matroids associated with such chains.

The other step towards to generalization in error correction is quasi-uniform codes [3]. It can be shown that for small length (n≤ 7 )  almost affine codes are linear and, therefore satisfy the Ingleton inequality. But there is exists a quasi-uniform code of length equal to 4 which violates the Ingleton inequality.

We continue with giving formal definitions of concepts discussed earlier, starting with almost affine codes.

Definition 1. Let F  be a finite set of cardinality q . A code CFN  is called almost affine if it satisfies the following condition for any subset X N

rXlogqCXN

As we can see F  from the definition above does not have to be a field. And the condition says that projection to any subset of [N]≔{1, 2, …,n}  has an integer dimension. It is easy to verify that any linear code C  satisfies this condition, therefore C  is an almost affine code and r  is called rank function of the code.

The main tool to study linear codes is their matrix representation. For almost affine codes we do not have such tool. But we have generalization of matrices – matroids. There are at least four equivalent definitions of matroids, we proceed with the definition via rank function.

Definition 2. Let E  be a finite set called ground set, and r  be a function r:2EN . Then matroid M  is a pair (E, r)  if r  satisfies the following axioms for any X E  and any x, y E

(R1 r=0 ,

(R2 rXrX{x}rX+1 ,

(R3 )  If  rX{x}=rX{y}=rX , then rX{x,y}=rX.

It can be shown that the function from definition 1 satisfies the axioms above, and an almost affine code C  induces a matroid. If r  from definition above satisfies only (R1 ) and (R2 ) axioms, then a pair E,r  is a demi-matroid. We continue with the core theorem of the article.

Theorem 1. Let CmCm-1⊂⋯⊂C1  be a chain of almost affine codes with respective rank functions rmrm-1r1 . Then E,ρm  with ρm=i=1m-1i+1ri  is a demi-matroid.

The prove of this theorem can be found in [6].

Demi-matroids have two types of duality. The dual demi-matroid to a given demi-matroid M=E,r  is  M*=E,r* , where r*Xr*X=X+rEX-rE . The supplement dual demi-matroid to a given one M=E,r  is M=E,r , where  rXrE-rEX . It is known fact due to [10], that M*=E,r* , M=E,r  and combination of the two types M*=E,r*  are demi-matroids.

Chains of almost affine codes

In this section we show that for any chain of almost affine codes CmCm-1⊂⋯⊂C1  and rank functions rmrm-1r1  pairs E,ηm , E,θm  and E,πm  are also demi-matroids with

ηmrm*-rm-1*++-1m+1r1*

θmr1-r2++-1m+1rm

πmrm*-rm-1*++-1m+1r1*

To do so we look at these three functions individually and show that under some circumstances they are equal to ρ* , ρ , ρ*   or just ρ .

Lemma 1. Let CmCm-1⊂⋯⊂C1  be a chain of almost affine codes with respective rank functions rmrm-1r1  and ρmr1-r2++-1m+1rm  and ηmrm*-rm-1*++-1m+1r1* . Then ηm=ρm*  if m  is odd and ηm=ρm   if m  is even.

Proof. For any nN0  we need to show

(*)  η2n=ρ2nη2n+1=ρ2n+1*

The equalities hold with n=0 . Assume (*) holds for any in , prove the induction step and show that η2n+2=ρ2n+2  and η2n+3=ρ2n+3*.  By assumption we have η2n+1=ρ2n+1* , notice that η2n+2=r2n+2*-η2n+1  by definition. Then η2n+2=r2n+2*-ρ2n+1* , so by the definition of the dual we have:

η2n+2X=r2n+2*X-ρ2n+1*X=X+r2n+2E-X-r2n+2E-X+ρ2n+1E-X-ρ2n+1E==ρ2n+1-r2n+2E-ρ2n+1-r2n+2E-X=ρ2n+2X

Now we look at η2n+3=r2n+3*-η2n+2 , combining with the above we have η2n+3=r2n+3*-ρ2n+2 , so by the definition of the supplement dual we have:

η2n+3X=r2n+3*X-ρ2n+2X=X+r2n+3E-X-r2n+3E-ρ2n+2E-ρ2n+2E-X==X+ρ2n+2+r2n+3E-X-ρ2n+2+r2n+3E=ρ2n+3*X.

Thus, lemma is proven by induction.

Lemma 2. Let CmCm-1⊂⋯⊂C1  be a chain of almost affine codes with respective rank functions rmrm-1r1  and ρmr1-r2++-1m+1rm  and θmr1-r2++-1m+1rm , then θm=ρm .

Proof.

θmX=i=1m-1i+1ri=i=1m-1i+1riE-riEX==i=1m-1i+1riE-i=1m-1i+1riEX=ρmX

Before the last lemma we need the fact that X E  we have r*X=r*X=X-rX . Indeed, by definition, we have

r*X=r*E-r*E-X=E+r-rE-E-X+rX-rE=|X|-r(X)

At the same time, we have

r*X=X+rE-X-rE=X+rE+rX-rE+r=X-rX.

Lemma 3. Let CmCm-1⊂⋯⊂C1  be a chain of almost affine codes with respective rank functions rmrm-1r1  and ρmr1-r2++-1m+1rm  and πmrm*-rm-1*++-1m+1r1* . Then πm=ρm  if m  is even and πm=ρm*   if m  is odd.

Proof. The proof is similar to the proof of lemma 1.

These three lemmas lead us to the following theorem.

Theorem 2. Let CmCm-1⊂⋯⊂C1  be a chain of almost affine codes with respective rank functions rmrm-1r1 , then the pairs E,ηm , E,θm  and E,πm  are demi-matroids.

Proof. By the theorem 1, M=E,ρm  is a demi-matroid, M*=E,ρm* , M=E,ρm  and M*=E,ρm*  are demi-matroids as well. And by the previous lemmas we showed that the pairs E,ηm , E,θm  and E,πm  are equal to E,ρm* , E,ρm , E,ρm*  or E,ρm  depending on the parity of m . Hence, the theorem is proven.

The theorem above is also proven in the article [6], but this prove is different. Applications of duality of chains of almost affine codes can be found in [6], [7] and [10].

Quasi-uniform codes

For proper introduction to quasi-uniform codes one can look at the article [3], where authors introduced these codes via random variable vectors. The simple recap is that quasi-uniform code CZE  induces a random variable vector which is distributed uniformly over projection to any X E . This class of codes does not have rank function r  which can be used to construct its matroid. But quasi-uniform codes induce a polymatroids instead.

Definition 3. For a finite set E  and h  be a real value function h :2ER , then the pair (E, h)  is a polymatroid if h  for any A, BE  satisfies the following axioms:

(R1 h=0 ,

(R2 AB ⇒hAhB ,

(R3 hAB+hA BhA+hB.

If h  satisfies cardinality bound hAE  and hA  is a non-negative integer, then (E, h)  is a matroid. If h  from definition above satisfies only (R1 ) and (R2 ) axioms, then a pair (E,h)  is a demi-polymatroid.

Any given quasi-uniform code C  induces a polymatroid by defining hXHCX , where HCX  is the entropy function of the codeword random variable. We conjecture that the chain duality discussed in the previous section holds for chains of quasi-uniform codes with respective demi-polymatroids. Recently it was shown in [8] that similar chain duality holds for rank-metric codes with (n, m) -demi-polymatroids which are the special case of demi-polymatroids. However, the general case is still open.

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