A partition of the hypercube into maximally nonparallel Hamming codes
aa r X i v : . [ c s . I T ] A p r A partition of the hypercubeinto maximally nonparallel Hamming codes ∗ Denis S. Krotov † Abstract
By using the Gold map, we construct a partition of the hypercubeinto cosets of Hamming codes such that for every two cosets the cor-responding Hamming codes are maximally nonparallel, that is, theirintersection cardinality is as small as possible to admit nonintersectingcosets.
Partitions of the hypercube into 1-perfect codes, especially into 1-perfectcodes that are not translations of each other, attract some attention in liter-ature [9, 15, 20, 21]. Such partitions themselves are perfect codes in a mixedalphabet, and they are used in the construction of binary perfect codes. Inthis note, we apply crooked permutations in order to construct a class ofsuch partitions with extremal properties. The partitions constructed consistof cosets of linear codes. On the other hand, such a partition is, in somesense, as far as possible from being ‘linear’: the affine span of every two ofits codes coincides with the whole space. The result of the note provides onemore example of application of the crooked functions in the construction ofextremal combinatorial structures. In the main part of the note (Section 2and 3) we use the Gold functions for construction. Section 2 describes theconstruction, and in Section 3 the symmetries of the constructed partitions ∗ This is the peer reviewed version of the following article: [D. S. Krotov.A partition of the hypercube into maximally nonparallel Hamming codes. J.Comb. Des. 24(4) 2014, 179-187], which has been published in final form at http://dx.doi.org/10.1002/jcd.21363 . This article may be used for non-commercialpurposes in accordance with Wiley Terms and Conditions for Self-Archiving. † Sobolev Institute of Mathematics, Novosibirsk, Russia; Novosibirsk State University,Novosibirsk, Russia. E-mail: [email protected]
Survey: -perfect partitions There are several works on constructions of 1-perfect partitions (partitionsof the hypercube into 1-perfect codes, i.e., codes with parameters of theHamming code) and some papers where such partitions are used in the con-struction of 1-perfect codes.The first construction of 1-perfect codes that uses 1-perfect partitionswas proposed by Solov’eva in [19] and, independently, by Phelps in [13]. Theconstruction uses two 1-perfect partitions of length n to construct a 1-perfectcode of length 2 n + 1. In the subsequent paper [14], Phelps generalized theconstruction: 2 k k ( n + 1) −
1. It should be noted that the earlier constructionby Heden [8], which was formulated in terms of codes over mixed alphabetinstead of partitions, is equivalent to the Solov’eva construction with somerestriction: one of the partitions consists of translations of the same code.The constructions of Solov’eva [19] and Phelps [14] exploit the principlesof the X N n anda 1-perfect code of length N over the ( n + 1)-ary alphabet (while 1-perfectcodes over non-binary alphabets are not considered in this paper, this isthe only place where they are mentioned) can be combined to construct a1-perfect code of length nN .In [19], Solov’eva proposed two constructions of 1-perfect partitions, oneof them (an analog of the known Vasil’ev construction of 1-perfect codes)giving at least 2 ( n − / different partitions.Rif`a and Vardy [18] proved that for every 1-perfect code, there are nonequiv-alent 1-perfect partitions that contain this code as a cell.Rif`a [16] constructed 1-perfect partitions in an algebraic way, extendingSteiner triple systems (STS) that satisfy some special properties, so-calledwell-ordered STS. Rif`a, Pujol and Borges in [17], Borges, Fernandez, Rif`aand Villanueva in [5] established connections between some combinatorial2nd metrical properties of a 1-perfect partition and algebraic properties ofits codes.In [15], Phelps enumerated all 11 nonequivalent 1-perfect partitions ofthe 7-cube and used them to evaluate the number of nonequivalent 1-perfectcodes of length 15 (the exact number was found nine years later by ¨Osterg˚ardand Pottonen [12]).In [1], Avgustinovich, Lobstein, and Solov’eva investigated the arrays ofthe cardinalities of the pairwise intersections of the cells of two 1-perfectpartitions.In [2], by Avgustinovich, Solov’eva, and Heden, partitions of the spaceinto mutually nonequivalent 1-perfect codes are constructed for every length2 m − ≥ { C , . . . , C n } definedas min i,j,i = j (dim h C i ∪ C j i − dim C i )do not increase with the growth of the length n . For the partition constructedin Section 2, this index has the order log n , which implies an essential im-provement in the direction of ‘non-parallelity’. Let m ≥ F be the finite field GF(2 m ) of order 2 m . Let σ be apower of 2, and assume that σ ± m − s, m ) = 1, where σ = 2 s .For example, σ = 2. Since the powers of a primitive element generating F ∗ = F \{ } are calculated modulo 2 m −
1, the condition gcd( σ ± , m −
1) = 1means that both x → x σ +1 , which is known as the Gold map, and x → x σ − are one-to-one mappings. We will treat the codes C of length 2 m (or 2 m − F ( F ∗ = F \ { } , respectively), i.e., C ⊂ F C ⊂ F ∗ ). The code distance is the minimal cardinality of the symmetricdifference X △ Y = ( X \ Y ) ∪ ( Y \ X ) of two different elements X , Y of thecode. A code is linear if it is a subspace of the vector space (2 F , △ , · ) overGF(2) (the set 2 F of all subsets of F is endowed with the addition △ andthe natural multiplication by a scalar: 1 · X = X , 0 · X = ∅ ). A Hammingcode ( extended Hamming code ) is defined as a collection of subsets X of F ∗ (even-cardinality subsets of F , respectively) satisfying P x ∈ X π ( x ) = 0 where π is some fixed permutation of F ∗ ( F , respectively).Recall some facts:(A) for all x, y ∈ F : ( x + y ) σ = x σ + y σ (derived from ( x + y ) = x + y );(B) for all x ∈ F : x σ + x + 1 = 0 (indeed, otherwise ( x + 1) σ +1 = ( x +1)( x + 1) σ = ( x + 1)( x σ + 1) = x σ +1 + x σ + x + 1 = x σ +1 , which isimpossible as f ( x ) = x σ +1 is one-to-one);(C) the cardinality of the code B = { X ∈ F : P x ∈ X , P x ∈ X x =0 , P x ∈ X x σ +1 = 0 } is 2 m − m − (in fact, B is a linear code of distance atleast 6, which has the maximal cardinality among the linear distance-6codes of length 2 m [6]; in the case σ = 2, a BCH code).For α ∈ F , p ∈ { , } , define the code H pα as the collection of subsets X of F satisfying X x ∈ X p, X x ∈ X ( x + α ) σ +1 = 0 . Clearly, H α is an extended Hamming code, and H α is a coset of H α . Theorem. (i)
The codes H α , α ∈ F , are mutually disjoint. (ii) The inter-section of two different codes H α and H β has the cardinality m − m . Proof.
An element X of the intersection of H pα and H pβ satisfies X x + α ∈ X p, (1) X x + α ∈ X x σ +1 = 0 , (2) X x + α ∈ X ( x + α + β ) σ +1 = 0 . (3)4e derive X x + α ∈ X ( x + α + β ) σ +1 = X x + α ∈ X ( x + ( α + β )) σ ( x + ( α + β )) (A) = X x + α ∈ X x σ +1 + X x + α ∈ X x σ ( α + β ) + X x + α ∈ X x ( α + β ) σ + X x + α ∈ X ( α + β ) σ +1( )(A)( ) = (cid:16) X x + α ∈ X x (cid:17) σ ( α + β ) + (cid:16) X x + α ∈ X x (cid:17) ( α + β ) σ + p ( α + β ) σ +1 . For p = 1 and α = β , the last expression cannot be equal to 0, by (B), whichcontradicts (3) and proves (i).For p = 0, the last expression above is equal to (cid:16) X x + α ∈ X x (cid:17)(cid:18)(cid:16) X x + α ∈ X x (cid:17) σ − + ( α + β ) σ − (cid:19) ( α + β ) , which implies, together with (3) and α = β , thateither X x + α ∈ X x = 0 or X x + α ∈ X x = α + β. By (C), each of these two cases, together with (1) and (2), has exactly2 m − m − solutions for X . So, there are 2 m − m solutions in total, whichproves (ii). N As a corollary , we partitioned all the odd-cardinality subsets of F into 2 m cosets H α of extended Hamming codes such that every two different cosetsare maximally nonparallel. Being maximally nonparallel here means thatthe intersection of the corresponding extended Hamming codes is as small aspossible to admit disjoint odd cosets. Note that, in general, the dimensionof the intersection of two Hamming codes (extended Hamming codes) canpossess any value from 2 m − m − m − m − F ), which means that the cosets of these codes (oddcosets, for extended Hamming codes) necessarily intersect.By removing the zero element from all X , we obtain a partition of the(2 m − m = 3) partition is equivalent to Partition 8 in the classification of all par-titions of the 7-cube into 1-perfect codes given by Phelps in [15, Appendix]. Remark.
We can consider α as the “color” of the elements of H α . Itis easy to see that, given an odd-cardinality set X ⊂ F , its color can be5xplicitly calculated by the formula α = α ( X ) = X x ∈ X x + (cid:16) X x,y ∈ X, x = y x σ y (cid:17) / ( σ +1) . Let us consider some isometries of the space that stabilize the constructedpartition H = { H α } α ∈ F of the odd-cardinality subsets of F . For convenience,define H α ( β ), α, β ∈ F , as the set of odd-cardinality subsets X ⊂ F satisfying X x ∈ X ( x + α ) σ +1 = β σ +1 (4)(in particular, H α (0) = H α ), and define H as the set of even-cardinalitysubsets Y ⊂ F satisfying X x ∈ Y x = 0 (5)(recall that H is an extended Hamming code; so, every odd-cardinality subsetof F is at distance one from exactly one element of H ).Direct verifications confirm the validity of the following four statements: Lemma 1.
For every δ from F , the permutation x → x + δ of F maps H α ( β ) to H α + δ ( β ) . Proof. If X meets (4), then Y = X + δ satisfies P x ∈ Y + δ ( x + α ) σ +1 = β σ +1 , which is equivalent to P y ∈ Y ( y + ( α + δ )) σ +1 = β σ +1 . N Lemma 2.
For every µ from F ∗ , the permutation x → µx of F maps H α ( β ) to H µα ( µβ ) . Proof. If X meets (4), then Y = µX satisfies P x ∈ µ − Y ( x + α ) σ +1 = β σ +1 . Replacing x = µ − y , we get P y ∈ Y ( µ − y + α ) σ +1 = β σ +1 , and, multi-plying by µ σ +1 , we obtain P y ∈ Y ( y + µα ) σ +1 = ( µβ ) σ +1 . N Lemma 3.
The automorphism x → x of the field F maps H α ( β ) to H α ( β ) . Lemma 4.
For every Y from H , the mapping X → X △ Y maps H α ( β ) to H α (( β σ +1 + s Y ) / ( σ +1) ) where s Y = P x ∈ Y x σ +1 . In particular, if s Y = 0 , themapping maps H α to itself. Proof.
From (4) we have P x ∈ X △ Y ( x + α ) σ +1 = P x ∈ Y ( x + α ) σ +1 + β σ +1 = P x ∈ Y x σ +1 + α (cid:0)P x ∈ Y x (cid:1) σ + α σ P x ∈ Y x + α σ +1 P x ∈ Y β σ +1 = s Y + β σ +1 , where the last equality comes from (5) and from the even cardinality of Y . N
6y an automorphism of the partition H we mean a transform X → π ( X ) △ Y , where π is a permutation of F and Y is a fixed subset of F , thatmaps every cell of H to another cell of H . The partition is called vertex-transitive if the automorphism group acts transitively on the vertices, theodd-cardinality subsets of F . That is, for every two odd-cardinality subsets X , Y of F there is an automorphism of H that sends X to Y . Similarly, the cell transitivity is defined (see [20] for examples of cell-transitive partitions),which property is, evidently, weaker than the vertex transitivity. The parti-tion is called 2 -cell-transitive if the automorphism group acts transitively onthe ordered pairs of codes from H ; that is, for every different H ′ , H ′′ from H and every different H ′′′ , H ′′′′ from H there is an automorphism of H thatmaps H ′ to H ′′′ and H ′′ to H ′′′′ . Proposition 1.
The partition H of the odd-cardinality subsets of F is -cell-transitive. Proof.
By Lemmas 1 and 2, for every different δ and α from F , themapping M δ,α : x → ( α + δ ) − ( x + δ ) maps H δ and H α to H and H ,respectively. N Proposition 2.
Under the action of the automorphism group of H , the setof odd-cardinality subsets of F is partitioned into at most two orbits. Proof.
Since H is an extended Hamming code, every odd-cardinalitysubset X of F is at distance 1 from some Y = Y ( X ) ∈ H .Let us show that X and Z such that s Y ( X ) , s Y ( Z ) = 0 (in notation ofLemma 4) belong to the same orbit. By Lemma 4, the mapping M X : X → X △ Y maps every H α , α ∈ F , to H α ( β ) with β = s / ( σ +1) Y . More-over, X is mapped to { δ } for some δ . By Lemma 2, the mapping N X : x → β − x maps H α ( β ) to H β − α (1), while N X ( M X ( X )) = N X ( { δ } ) = { β − δ } . Finally, by Lemma 1, the mapping L X : x → x + β − δ maps H β − α (1) to H β − α + β − δ (1), while L X ( N X ( M X ( X ))) = { } . So, the mapping L X ( N X ( M X ( · ))) maps H to { H α (1) } α ∈ F and sends X to { } . The sameis true for Z . In summary, M − Z ( N − Z ( L − Z ( L X ( N X ( M X ( X )))))) = Z and M − Z ( N − Z ( L − Z ( L X ( N X ( M X ( · )))))) is an automorphism of H . That is, X and Z are from the same orbit.Similar (even more simple) arguments solve the case s Y ( X ) = s Y ( Z ) = 0. N A computer experiment shows that for m = 5 , , , ,
13 the partition weconstruct is not vertex-transitive (for m = 3, it is); i.e., the number of theorbits is exactly 2. An invariant that distinguishes the vertices of differentorbits is the number of two-color squares in the neighborhood: for a given7ertex X , we count the number Q X of quadruples { X △{ x, y } , X △{ y, z } , X △{ z, v } , X △{ v, x }} such that α ( X △{ x, y } ) = α ( X △{ z, v } ), and α ( X △{ y, z } ) = α ( X △{ v, x } ).It occurs that for two vertices X = ∅ and Y from different orbits, the numbers Q X and Q Y are different, m ≤
13. Moreover, they depend on σ , whichimplies that the construction, for fixed m = 5 , , , ,
13, gives nonequivalentpartitions. Here is the list of the calculated tuples (2 m , σ + 1 , Q X , Q Y ): ( 2 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , (it is sufficient to consider only the cases σ < m/ , as the partitions for σ = 2 s and for σ = 2 m − s coincide, which easily follows from the identity x s +1 = ( x m − s ) s ). Observations. 1.
For σ = 2 ( m − / , we have got Q X = 0. The value Q Y is rather close to the “average” value D/ (2 m −
1) where D = (2 m − m − − m − is the number of the pairs { X △{ x, y } , X △{ z, v }} such that α ( X △{ x, y } ) = α ( X △{ z, v } ). A two-color square have never been extended to a three-color octahedron;i.e., α ( X △{ x, z } ) = α ( X △{ y, v } ), in the notations above. The length-2 partitions with σ + 1 = 3 and σ + 1 = 5 are nonequivalent(while ( Q X , Q Y ) coincide). This fact was established using the non-equivalence ofthe so-called local quasigroups( F, • ) : x • y = α ( { }△{ x }△{ y } )of the two partitions. The calculations were made using the isomorphism searchfunctionality of the LOOPS package [11] for GAP [22].An evident conjecture is that the number of orbits is 2 for every odd m ≥ H is a composition of automorphismsfrom Lemmas 1, 2, 3, and 4 (with s Y = 0). For σ = 2, this follows from [4] (anyautomorphism of H is an automorphism of the code { Y ∈ H : s Y = 0 } , which isan extended double-error-correcting BCH code). As follows from the proof, the claim (i) of the theorem will remain valid if wereplace the function f ( x ) = x σ +1 by an arbitrary permutation (bijective function) : F → F such that for all α = 0 the set H α = { f ( x ) + f ( x + α ) : x ∈ F } is anaffine subspace (over GF(2)) of F (the sum of an odd number of elements of H α belongs to H α , which does not contain 0 as f is bijective). A class of functionsthat obviously satisfy this condition is the class of quadratic function, i.e., thevector functions whose components are represented as polynomials of degree atmost 2. For such permutations, the cardinalities of the mutual intersections of thecorresponding Hamming codes can be counted in terms of the cardinalities of H α ,using the technique from the second part of the theorem. Another appropriateclass of permutations is the following. A permutations f : F → F is called crooked [3] if for all α = 0 the set H α is an affine hyperplane. The Gold functions arecrooked and quadratic; at the moment, all known crooked functions are quadratic.Does (ii) hold for the non-quadratic crooked permutations f ( x ) (if there are some)?See [23] for other applications of the crooked permutations in the construction ofdifferent extremal combinatorial structures. Do there exist partitions of the hypercube into cosets of maximally non-parallel Hamming codes for even m ? In [9], partitions of the space into mutuallynonparallel cosets of Hamming codes are constructed for all m ≥
3. The prob-lem of minimizing mutual intersection of the Hamming codes for such a partitionremains open for even m . Acknowledgments
The author thanks Sergey Avgustinovich for discussions, Gohar Kyureghyan forconsulting in the area of crooked functions, and the referees for many suggestions,which greatly improved the presentation. The research was partially supportedby the RFBR (grant 13-01-00463), by the Ministry of education and science ofRussian Federation (project 8227), and by the Target program of SB RAS for2012-2014 (integration project No. 14).
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