Non-linear Cyclic Codes that Attain the Gilbert-Varshamov Bound
Ishay Haviv, Michael Langberg, Moshe Schwartz, Eitan Yaakobi
aa r X i v : . [ c s . I T ] J a n Non-linear Cyclic Codes that Attain theGilbert-Varshamov Bound
Ishay Haviv,
Member, IEEE , Michael Langberg,
Senior Member, IEEE , Moshe Schwartz,
SeniorMember, IEEE , and Eitan Yaakobi
Member, IEEE
Abstract —We prove that there exist non-linear binary cycliccodes that attain the Gilbert-Varshamov bound.
Index Terms —Cyclic codes, non-linear codes, Gilbert-Varshamov bound, good family of codes
I. I
NTRODUCTION
For a finite field F q , a cyclic code of length n is a linearsubspace C ⊆ F nq that is closed under cyclic permutations, i.e.,for every codeword x ∈ C , all cyclic permutations of x arealso included in C . Cyclic codes have been extensively studiedover the last decades exhibiting a rich algebraic structure withimmense applications in storage and communication, e.g., [3],[7], [11], [13].A code over alphabet Σ is said to have (normalized)minimum distance at least δ ∈ [
0, 1 ] if any two distinctcodewords in C ⊆ Σ n are of Hamming distance at least δ n .It is also said to have rate R ( C ) , n log | Σ | | C | . Of particularinterest are good families of codes , which are sequences ofcodes C , C , . . . , with C i of length n i , rate R i , and minimumnormalized distance δ i , such that simultaneously, lim i → ∞ n i = ∞ , lim i → ∞ R i > and lim i → ∞ δ i > One of the most fundamental challenges in coding theoryis to construct good families of codes. Several such familieswere presented in the literature over the years. This includes,for example, the Gilbert-Varshamov (GV) codes [8], [15] andthe algebraic constructions of Justesen [10] and Goppa [9] (seealso [11]). However, the existence of a good family of cyclic (linear) codes is a long-standing open problem, see e.g., [2],[5], [6], [12].In an attempt to shed some light on the problem of goodcyclic codes, some works considered close variants to cycliccodes. Quasi-cyclic and double-circulant codes are one exam-ple, created by interleaving cyclic codes. These families wereshown to contain families of good codes [11, Ch. 16]. Another
Ishay Haviv is with the School of Computer Science at The AcademicCollege of Tel Aviv-Yaffo, Tel Aviv 61083, Israel.Michael Langberg is with the Department of Electrical Engineering, StateUniversity of New-York at Buffalo, Buffalo, NY 14260, USA (e-mail:[email protected]).Moshe Schwartz is with the Department of Electrical and ComputerEngineering, Ben-Gurion University of the Negev, Beer Sheva 8410501, Israel(e-mail: [email protected]).Eitan Yaakobi is with the Department of Computer Science, Tech-nion – Israel Institute of Technology, Haifa 32000, Israel (e-mail:[email protected]).Authors appear in alphabetical order. Research supported in part by NSFgrant 1526771, the Israel Science Foundation (ISF) grants No. 130/14, andNo. 1624/14. such example is the family of module skew codes, which arealmost cyclic except for a slight twist in the permutation. Thesewere recently shown to contain a good family of codes [1].In this work, we study another variant of cyclic codes – non-linear cyclic codes . For simplicity of presentation weconsider only the binary case. We will show that this familyof codes contains a good family which asymptotically meetsthe GV bound [8], [15], i.e., of normalized distance δ andasymptotic rate approaching − H ( δ ) , where H stands for thebinary entropy function. This matches the best known lowerbound for binary codes of minimum normalized distance δ that are not necessarily cyclic. To the best of our knowledge,good families of non-linear binary cyclic codes have not beenpreviously presented in the literature.Our construction of good binary non-linear cyclic codes isconceptually very simple and includes two steps. In the firststep we construct a high-rate binary code which we call auto-cyclic . (All formal definitions are given below in Section II.)An auto-cyclic code is a non-linear cyclic code in whichthe set of cyclic permutations of any given codeword is of(normalized) minimum distance at least δ . In this context,we refer to δ as the auto-cyclic distance. Auto-cyclic codesare reminiscent of orthogonal or low auto-correlation codes,e.g. [4], [14]. Using a probabilistic argument, we show theexistence of a subset of {
0, 1 } n of asymptotic rate which isauto-cyclic with auto-cyclic distance δ arbitrarily close to .Once a high-rate auto-cyclic code is established, we greedilyremove some of its elements (using a slight variant of the wellknown greedy process that leads to the GV bound) to obtainthe desired non-linear cyclic code C of rate − H ( δ ) andminimum distance δ .The remainder of this note is structured as follows. InSection II we present our formal definitions. In Section IIIwe prove the existence of binary non-linear cyclic codes thatmeet the GV bound. Section IV includes concluding remarksand open questions. II. P RELIMINARIES
Let [ n ] , {
0, 1, . . . , n − } . In the context of indices, alladdition and multiplication operations are done modulo n . Definition 1.
Let x = x , . . . , x n − ∈ {
0, 1 } n . Forall i ∈ [ n ] ,thecyclicshiftof x i -locationstotheleftisdefinedas E i ( x ) , x i , . . . , x n − , x , . . . , x i − . It is common in the literature to define cyclic codes as linear. Thus, weshall make it a point to emphasize the fact that the codes we consider maybe non-linear by naming them non-linear cyclic codes . Definition 2.
Let x = x , . . . , x n − , y = y , . . . , y n − ∈{
0, 1 } n . The (normalized) Hamming distance between x and y isdefinedas d ( x , y ) , |{ i ∈ [ n ] : x i = y i }| n . ThecyclicHamming-distancebetween x and y isdefinedas d cyc ( x , y ) , min i ∈ [ n ] d ( E i ( x ) , y ) = min i ∈ [ n ] d ( x , E i ( y )) . The auto-cyclic Hamming-distance between x and itself isdefinedas d ∗ cyc ( x , x ) , min i : E i ( x ) = x d ( E i ( x ) , x ) . Noticethatinthedefinitionoftheauto-cyclicdistanceweonlyconsidershifts E i ( x ) thatdifferfrom x . Fortheall-0vector n andtheall-1 vector n wedefinetheauto-cyclicdistanceto be n . Definition 3.
Asubset C ⊆ {
0, 1 } n is cyclic ifforevery x ∈ C andevery i ∈ [ n ] itholdsthat E i ( x ) ∈ C . Definition 4.
Wesaythat C ⊆ {
0, 1 } n isan [ n , δ ] binary auto-cyclic code if C is cyclic and in addition for every x ∈ C itholdsthat d ∗ cyc ( x , x ) > δ . Definition 5.
We say that C ⊆ {
0, 1 } n is an [ n , δ ] binarynon-linear cyclic code if C iscyclicandinaddition,forevery x , y ∈ C , x = y ,itholdsthat d cyc ( x , y ) > δ . Definition 6.
Therateofasubset C ⊆ {
0, 1 } n isdefinedby R ( C ) , log | C | n . Definition 7.
The asymptotic rate of an infinite sequence ofcodes C = { C i } ∞ i = , where C i ⊆ {
0, 1 } n i , n i < n i + forall i ,isdefinedas R ( C ) , lim sup i → ∞ R ( C i ) = lim sup i → ∞ log | C i | n i . III. N ON - LINEAR C YCLIC C ODES
We start with the following lemma that provides a proba-bilistic construction of binary auto-cyclic codes of high rate.
Lemma 8.
For every δ < and every sufficiently largeprime n ,thereexistsan [ n , δ ] binaryauto-cycliccodeofrateatleast − n .Inparticular,thereexistsasequenceofbinaryauto-cycliccodesofnormalizedminimumdistance δ andasymptoticrate . Proof:
Let n be a sufficiently large prime. Considerchoosing a random element x = x , . . . , x n − in {
0, 1 } n from the uniform distribution. Namely, each entry of x ischosen i.i.d. uniformly over {
0, 1 } . We study the probabilitythat d ∗ cyc ( x , x ) > δ . Let i ∈ [ n ] \ { } . We first analyze theprobability Pr x h d ( E i ( x ) , x ) > δ i . In our analysis we will use the fact that for a prime n , thesequence i , 2 i , 3 i , . . . , ( n − ) i consists of elements that areall distinct modulo n . For k ∈ [ n ] , let A k be the indicator of the event that the ki -th coordinate (modulo n ) of x differsfrom that of E i ( x ) , namely that x ki differs from x ( k + ) i . As n is prime it holds that Pr x h d ( E i ( x ) , x ) > δ i = Pr x ∑ k ∈ [ n ] A k > δ n . Let a = a , . . . , a n − ∈ {
0, 1 } n be an arbitrary vector, andconsider the probability Pr [ ∀ k : A k = a k ]= n − ∏ k = Pr [ A k = a k | A = a , . . . , A k − = a k − ] . Notice that each event A k depends only on x ki and x ( k + ) i .Using the fact that the sequence i , 2 i , 3 i , . . . , ( n − ) i con-sists of elements that are all distinct modulo n , we concludethat for k ∈ {
0, . . . , n − } : Pr [ A k = a k | A = a , . . . , A k − = a k − ] =
12 .
Here, we used the fact that each entry x i of x is uniformover {
0, 1 } and that x ( k + ) i is independent of the entries { x ℓ i } ℓ ∈ [ k + ] that determine the events A , . . . , A k − . Thus, Pr [ ∀ k : A k = a k ] n − . We now conclude that for any i ∈ [ n ] \ { } , Pr x h d ( E i ( x ) , x ) < δ i = Pr x ∑ k ∈ [ n ] A k < δ n ∑ a ∈{ } n d ( a ,0 ) < δ Pr x [ ∀ k : A k = a k ] H ( δ ) n n − . Above, we upper bound the Hamming ball of radius δ n by H ( δ ) n , where H stands for the binary entropy function. Bythe union bound over i ∈ [ n ] \ { } , we have Pr x h d ∗ cyc ( x , x ) < δ i ( n − ) · ( H ( δ ) − ) n . (1)Finally, define C , n x ∈ {
0, 1 } n : d ∗ cyc ( x , x ) > δ o . By (1) we have | C | > n − ( n − ) · H ( δ ) n > n − , for δ < and a sufficiently large n , so the rate of C is atleast − n . In addition, if x ∈ C then any cyclic shift of x is also in C .Equipped with Lemma 8, we are ready to prove our mainresult stated below. Theorem 9.
For every < δ < and R < − H ( δ ) there exists a sequence of binary non-linear cyclic codes ofnormalizedminimumdistanceatleast δ andasymptoticrateatleast R . Proof:
Fix ε > and let R , − H ( δ ) − ε . We constructfor every sufficiently large prime n an [ n , δ ] binary non-linearcyclic code C of rate at least R . Our construction has twosteps. In the first step, using Lemma 8, we construct an [ n , δ ] auto-cyclic code C ′ of rate at least − ε .In the second step we construct C by a greedy proceduresimilar to that used in the standard GV bound. Specifically, westart with C = ∅ . Let x be any word in C ′ . Add x and all itscyclic shifts (cid:8) E i ( x ) (cid:9) i ∈ [ n ] to C and remove them from C ′ . Inaddition, remove from C ′ all words y for which d cyc ( x , y ) < δ . In this process, since d cyc ( x , y ) = min i d ( E i ( x ) , y ) , weremove at most n H ( δ ) n words from C ′ . Here, as before, weupper bound the Hamming ball of radius δ n by H ( δ ) n . Notethat for any y removed, we also remove all its cyclic shifts.We now continue in iterations, in each iteration we add anelement x ∈ C ′ and all its cyclic shifts to C and remove all y for which d cyc ( x , y ) < δ from C ′ (including x and its cyclicshifts). We continue in this fashion until C ′ is empty.It follows that the code C constructed above has size at least | C | > | C ′ | n · H ( δ ) n , and thus has rate at least R ( C ) > − H ( δ ) − ε − o ( ) > R . The code C is an [ n , δ ] binary non-linear cyclic code. Namely,by construction, for each x ∈ C and i ∈ [ n ] it holds that E i ( x ) ∈ C . Moreover, by the iterative procedure, any distinct x , x ′ ∈ C satisfy d ( x , x ′ ) > δ .IV. C ONCLUSION
In this work we proved the existence of a good family ofbinary non-linear cyclic codes of normalized distance δ andasymptotic rate − H ( δ ) (i.e., codes that meet the GV bound).The codes we construct are non-linear. Specifically, for δ < ,the code C ′ obtained in the first step of our construction is thecollection of all codewords x with auto-cyclic distance greateror equal to δ . This code is not linear. To see this, consider any x for which for all i ∈ [ n ] \ { } it holds that d ( x , E i ( x )) > δ + n . The proof of Lemma 8 shows the existence of severalsuch x . Consider now the codeword y = x + n − . Thecodeword y satisfies the slightly weaker condition that for all i ∈ [ n ] \ { } : d ( y , E i ( y )) > δ . So both x and y haveauto-cyclic distance at least δ however x + y = n − hasauto-cyclic distance of n . The second step of our construction,which removes elements from C ′ to obtain the cyclic code C ofdistance δ and rate arbitrarily close to − H ( δ ) , is greedy anddoes not necessarily yield linear codes. Whether the first stepof our construction can be refined using algebraic techniquesto yield a linear auto-cyclic code, or whether the second step ofour construction (assuming a linear C ′ ) can yield a linear code C , are intriguing problems left open in this work. Problemsthat, if solved, may shed light on the existence of good binarylinear cyclic codes.V. A CKNOWLEDGMENT
The authors would like to thank Alexander Barg for hisvaluable comments on an earlier draft of this work. R
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