Classification of Δ-divisible linear codes spanned by codewords of weight Δ

SPANNED BY CODEWORDS OF WEIGHT∆

MICHAEL KIERMAIER AND SASCHA KURZ

ABSTRACT. We classify allq-ary∆-divisible linear codes which are spanned by codewords of weight∆.

The basic building blocks are the simplex codes, and forq= 2additionally the first order Reed-Muller codes and the parity check codes. This generalizes a result of Pless and Sloane, where the binary self-orthogonal codes spanned by codewords of weight4have been classified, which is the caseq= 2and∆ = 4of our classification. As an application, we give an alternative proof of a theorem of Liu on binary∆-divisible codes of length4∆in the projective case.

1. INTRODUCTION

In this article, all codes will silently considered to be linear. An[n, k]q code Cis ak-dimensional subspace of then-dimensionalFq-vector spaceFⁿq. Agenerator matrixof a linear[n, k]q-codeC is a matrix whose rows form a basis ofC. The generator matrix is calledsystematicif it starts with ak-by-k unit matrix. Up to a permutation of the positions, every linear code has a systematic generator matrix.

Elementsc∈Care calledcodewordsandn=n(C)is called thelengthof the code. Thesupportof a codewordcis the number of coordinates with a non-zero entry, i.e.,supp(c) ={i∈ {1, . . . , n} |ci6= 0}.

The(Hamming-)weightwt(c)of a codeword is the cardinality# supp(c)of its support. A codeC is called∆-divisibleif the weight of every codeword is divisible by some integer∆ ≥1. Divisible codes are important in coding theory and have applications in finite geometry, for example, see the surveys [17]

and [6]. The classification of divisible codes is a hard problem and has been solved only in special cases.

Recent results into this direction can be found in [10,?, 1, 5, 7, 9].

Given an[n, k]_q codeC, the[n, n−k]_q codeC^⊥ =

x∈Fⁿq |x^Ty= 0∀y∈C is called theor- thogonal, ordualcode ofC. A code is calledself-orthogonalifC ⊆ C^⊥ and self-dual ifC = C^⊥. Any self-orthogonal binary code is2-divisible, and any4-divisible binary code is self-orthogonal. In [13, Th. 6.5], indecomposable self-orthogonal binary codes which are spanned by codewords of weight4 are completely characterized. Based on the property that self-orthogonal binary codes spanned by code- words of weight4are always4-divisible, we are going to generalize that result. For this, the property of self-orthogonality is replaced by divisibility, which is in the spirit of the generalization of the theorem of Gleason and Pierce (see [16, Sec. 6.1]) by Ward [18, Th. 2]. In fact, this was Ward’s original motivation for studying divisible codes.

We will prove the following characterization ofq-ary∆-divisible codes that are generated by code- words of weight∆.

Theorem 1. Let∆be a positive integer and letabe the largest integer such thatq^adivides∆. LetCbe aq-ary∆-divisible linear code that is spanned by codewords of weight∆. ThenCis isomorphic to the direct sum of codes of the following form, possibly extended by zero positions.

(i) The_qk−1^∆ -fold repetition of theq-ary simplex code of dimensionk∈ {1, . . . , a+ 1}.

In the binary caseq= 2additionally:

(ii) The₂k−2^∆ -fold repetition of the binary first order Reed-Muller code of dimensionk∈ {3, . . . , a+ 2}.

(iii) Fora≥1: The ^∆₂-fold repetition of the binary parity check code of dimensionk≥4.

1

Up to the order, the choice of the codes is uniquely determined byC.

We would like to point out the following border cases which are covered by Theorem 1.

(a) The zero code of lengthnis∆-divisible and spanned by an empty set of codewords of weight

∆for any positive integer∆. It is covered by Theorem 1 as an empty direct sum, extended byn zero positions.

(b) The value ofacan be0. This appears whenever ∆is coprime toq, including also the trivial situation ∆ = 1. Here, only direct sums of ∆-fold repeated simplex codes of dimension 1 appear, which are the same as the ordinary repetition codes of length∆. In the case∆ = 1this means thatCis the full Hamming space of some dimension, possibly extended by zero positions.

Further related work includes the classical result of Bonisoli characterizing constant weight codes [3]

and the generalization to two-weight codes where one of the weights is twice the other [8].

2. PRELIMINARIES

LetCbe aq-ary linear code. Ifdim(C) = 0,Cis called azero code. A position where all codewords of a linear codeChave a zero entry is called azero position. Equivalently, all generator matrices have a zero column at that position. A code without zero positions is calledfull-length. Zero positions are irrelevant for many aspects of coding theory, such that it is often enough to consider full-length codes.

Two vectorsv, woverFq are called(projectively) equivalent, denoted asv∼w, if there is a non-zero scalarλ∈F^×q withv=λw. The highest number of equivalent positions of a code is called itsmaximum multiplicity. A full-length code of maximum multiplicity≤1is calledprojective. The zero code of length 0is the unique code of maximum multiplicity0.

ForM ⊆Fⁿq, we definesupp(M) = S

c∈Msupp(c), wheresupp(∅) = ∅. The numberneff(C) :=

# supp(C)is called theeffective lengthofC. The effective length equals the number of non-zero posi- tions ofC. The codeCis full-length if and only if the effective length ofCcoincides with the ordinary length.

LetN ={1, . . . , n}andc∈Fⁿq. ForI⊆N, we denote the restriction ofcto the positions inIbycI, that isc_I = (c_i)_i∈I. The restriction of a[n, k]_q-codeCof lengthnis defined asC_I ={c_I |c∈C}. It is known as the codepuncturedinN\I. Forc∈C, the punctured codeC_N\supp(c)is called theresidual with respect toc. Its dimension is at mostk−1but may also be strictly less. The extremal situation that the residual is a zero code appears if and only ifsupp(c) = supp(C). Note that residuals preserve the properties full-length and projective.

ByAi(C)we denote the number of codewords of weightiinCand byBi(C)the number of code- words of weightiinC^⊥. Mostly, we will just writeAiandBi, whenever the codeCis clear from the context. We always haveA0 = B0 = 1. A code with only a single non-zero weight is calledconstant weight code. Furthermore, the codeC is full-length if and only ifB1 = 0, and it is projective if and only ifB2 =B1= 0. Theweight distributionofCis the sequence(Ai). Theweight enumeratorofC is the polynomialWC(x) = Pn

i=0Aixⁱ ∈ Z[x], and thehomogeneous weight enumeratorofC is the homogeneous polynomialWC(x, y) =Pn

i=0Aixⁿ⁻ⁱyⁱ∈Z[x, y].

The weight distributions(Ai)and(Bi)are related by the famous MacWilliams identities [11]. For all i∈ {0, . . . , n},

n−i

X

j=0

n−j i

A_j=q^k−i

i

X

j=0

n−j n−i

B_j.

A compact form of the MacWilliams identities is given by the polynomial equation W_C⊥(x, y) = 1

#CWC(x+ (q−1)y, x−y) involving the homogeneous weight enumerators.

There are several alternative forms of the MacWilliams equations, among them the (Pless) power moments, see [12]. For full-length binary codes, the first four Pless power moments are

n

X

i=1

A_i = 2^k−1, (1)

n

X

i=1

iAi = 2^k−1n, (2)

n

X

i=1

i²Ai = 2^k−2(n(n+ 1) + 2B2), (3)

n

X

i=1

i³Ai = 2^k−3(n²(n+ 3) + 6nB2−6B3). (4) Twoq-ary linear codes C, C⁰ of the same lengthnare calledisomorphic orequivalent, denoted as C ∼=C⁰, ifCcan be mapped toC⁰via an isometry of the ambient Hamming spaceFⁿq, which preserves theFq-linearity of codes. The group of these isometries is given by the semimonomial group onFⁿq. For binary codes, it reduces to the symmetric groupSnacting the positions. Furthermore, for twoq-ary linear codesC, C⁰, not necessarily of the same length, we writeC∼=0C⁰if the codesCandC⁰are isomorphic after removing zero positions. IfCandC⁰of the same length,C∼=0C⁰is equivalent toC∼=C⁰.

The direct sum of an[n, k]qcodeCand an[n⁰, k⁰]qcodeC⁰is the[n+n⁰, k+k⁰]qcode C⊕C⁰={(c1, . . . , c_n, c⁰₁, . . . , c⁰_n)|(c₁, . . . , c_n)∈C,(c⁰₁, . . . , c⁰_n)∈C⁰}.

IfGis a generator matrix ofCandG⁰a generator matrix ofC⁰, a generator matrix ofC⊕C⁰is given by(^G_0G⁰0). A code is calleddecomposable if it is isomorphic to a direct sum of two non-zero codes.

Otherwise, it is calledindecomposable. We note that indecomposable codes may have zero positions. All zero codes and all linear codes of dimension1are indecomposable. A codeCis decomposable if and only if there are non-zero subcodesD, D⁰withD+D⁰=Candsupp(D)∩supp(D⁰) =∅. Each full- length linear code is isomorphic to an essentially unique direct sum of indecomposable full-length linear codes. For the original proof in the binary case see [15, Th. 2] and for the general case [2, Th. 6.2.7].

To show that some code is indecomposable, the following lemma may be helpful. It is essentially [2, 6.2.13].

Lemma 2.1. LetG= (Ik A)be a systematic generator matrix, whereIkdenotes thek×kunit matrix.

We consider the set of non-zero entries of Aas the vertex set of a simple graphG, where two distinct vertices are connected by an edge if and only if they are entries in the same row or the same column ofA.

Then the code generated byGis indecomposable if and only ifGhas a connected component contain- ing entries from each row ofA.

For a codeCwe denote them-fold repetition ofCbym·C. A codeCis a∆-divisible[n, k]q-code if and only ifm·Cis anm∆-divisible[mn, k]q-code. Moreover,Cis indecomposable if and only ifm·C is indecomposable. For the weight enumerator we haveW_m·C(x) =WC(x^m).

In this article, we want to classify all∆-divisibleq-ary linear codes which are generated by codewords of weight ∆. The investigated property is invariant under forming direct sums and adding zero posi- tions, such that it suffices to restrict the classification to indecomposable full-length codes with the stated property. These codes turn out to be repetitions of members of the following three families.

(i) Theq-ary simplex codeSimq(k)of dimensionk ≥ 1. A generator matrix can be constructed column-wise by taking a set of projective representatives of the non-zero vectors inF^kq. In the geometric setting, the simplex codeSimq(k)is the set of all points contained in a projective space of algebraic dimensionk. Its parameters are[^q_q−1^k−1, k]qand the corresponding weight enumerator

is

W_Simq(k)(x) = 1 + (q^k−1)x^qk−1.

So it is a code of constant weight ∆ = q^k−1 and in particular, the code is ∆-divisible and spanned by codewords of weight∆. Moreover, the codeSim_q(k)is full-length and projective and by Lemma 2.1, it is indecomposable.

(ii) Theq-ary first order Reed-Muller codeRM_q(k)of dimension k ≥ 2. A generator matrix can be constructed in the following way: Take all vectors inF^k−1q as the columns of a matrix, and extend that matrix by an additional row consisting of the all-one vector. Stated slightly different, RMq(k)is the span of(q−1)·Simq(k−1), extended by a zero position, and a vector of weight q^k. In the geometric setting, the first order Reed-Muller code is the set of all points contained in an affine space of dimensionk. Its parameters are[q^k−1, k]q and the corresponding weight enumerator is

W_RMq(k)(x) = 1 + (q^k−q)x^{(q−1)qk−2}+ (q−1)x^qk−1.

So it is a∆-divisible code with∆ = q^k−2. The code RMq(k)is full-length and projective.

Using Lemma 2.1 it is checked to be indecomposable for(q, k)6= (2,2).¹

The first order Reed-Muller codes will only be needed in the binary caseq= 2. Here, all but a single non-zero word are of weight∆, soRM2(k)is spanned by codewords of weight∆.

(iii) Theq-ary parity check codePCq(k)of dimensionk≥ 1. It is the set of all vectorsc ∈F^k+1q

withc1+. . .+ck+1 = 0. A generator matrix can be constructed column-wise by taking the kstandard basis vectors inF^kq together with the negated all-one vector. Geometrically,PC_q(k) corresponds to a projective basis of a projective space of algebraic dimensionk, sometimes also called projective frame. Its parameters are[k+ 1, k]_q. It is the dual code of the(k+ 1)-fold q-ary repetition code, so the polynomial form of MacWilliams gives the homogeneous weight enumerator as

W_PCq(k)(x, y) = 1

q (x+ (q−1)y)^k+1+ (q−1)(x−y)^k+1 .

The codePCq(k)is full-length and indecomposable by Lemma 2.1. Moreover,PCq(k)is pro- jective if and only ifk6= 1.

The parity check code will only be needed in the binary caseq = 2. Here,PC₂(k)is just the set of all vectors inFⁿ⁺¹2 of even weight. Thus it is a∆-divisible code with∆ = 2, and the above stated generator matrix shows thatPC2(k)is spanned by codewords of weight∆.

Remark2.2. There are the following isomorphisms among these series of codes.

PCq(1)∼= 2·Simq(1), PC2(2)∼= Sim2(2) , PC2(3)∼= RM2(3) and PC3(2)∼= RM3(2).

Up to taking repetitions, the above stated properties show that there are no other isomorphisms among possibly repeated codes of these series.

We will need the following three lemmas on divisible codes. The first one is known as the Theorem of Bonisoli [3], which says that every constant-weight code is the repetition of some simplex code, possibly extended by zero positions. We state the theorem it in slightly more refined way.

Lemma 2.3. Letqbe a prime power and∆a positive integer. Letabe the largest integer such thatq^a divides∆.

LetC be aq-ary linear code of dimensionk ≥ 1of constant weight∆. Thenk ≤ a+ 1and C is isomorphic to the _qk−1^∆ -fold repetition of theq-ary simplex codeSim_q(k), possibly extended by zero positions.

1In the border case(q, k) = (2,2), the first order Reed-Muller code is just the full Hamming spaceF²2, which is decomposable.

The second lemma shows that it is enough to considerF^q-linear∆-divisible codes where∆is a power of the base prime. It is essentially [18, Th. 1], see also [19, Th. 2].

Lemma 2.4. Letq= p^rbe a prime power withpprime and∆a positive integer. Letabe the largest integer withp^a |∆.

LetCbe aFq-linear∆-divisible code. ThenCis isomorphic to the_p^∆a-fold repetition of ap^a-divisible Fq-linear code, possibly extended by zero positions.

The third lemma is the result [20, Lemma 13] on the divisibility of residual codes, see also [6, Lemma 7].

Lemma 2.5. Letqbe a prime power withpand∆a positive integer.

LetCbe aq-ary∆-divisible code. Then any residual ofCis _gcd(q,∆)^∆ -divisible.

3. THE CHARACTERIZATION

In this section, we will prove Theorem 1.

Lemma 3.1. LetCbe a∆-divisibleq-ary linear code andc, c⁰ ∈Ctwo codewords of weight∆. Then one of the following statements is true:

(i) suppc= suppc⁰andc∼c⁰.

(ii) #(suppc∩suppc⁰) = ^q−1_q ∆and#{i∈suppc∩suppc⁰ |c⁰_i =λci}= ¹_q∆for allλ∈F^∗q. (iii) suppc∩suppc⁰=∅.

Proof. Letb= #(suppc∩suppc⁰)anda_λ= #{i∈suppc∩suppc⁰|c_i=λc⁰_i}forλ∈F^∗q. Then X

λ∈F^∗q

aλ=b. (5)

Letλ∈F^∗q. We have

w(c−λc⁰) = #(suppc\suppc⁰) + #(suppc⁰\suppc) + #{i∈supp(c)∩supp(c⁰)|c_i6=λc⁰_i}

= (w(c)−b) + (w(c⁰)−b) + (b−a_λ)

= 2∆−b−aλ.

If suppc = suppc⁰, we have w(c −λc⁰) = ∆−a_λ. By the ∆-divisibility, a_λ ∈ {0,∆}, and Equation (5) shows thata_λ = ∆for a uniqueλ∈F^∗q. Thereforec =λc⁰ for this value ofλ, which is case (i).

Assuming that we are not in case (i) or (iii), the above expression forw(c−λc⁰)is neither0nor2∆.

By the∆-divisibility ofC, the only remaining possibility isw(c⁰ −λc) = ∆. Soa := a_λ = ∆−b independently ofλ ∈ F^∗q. Now Equation (5) yieldsb = ^q−1_q ∆anda = ¹_q∆, showing that we are in

case (ii).

For the inductive treatment of indecomposable codes, we prepare the following lemma.

Lemma 3.2. LetCbe an indecomposable linear code of dimensionkandBa basis ofCconsisting of codewords of minimum weight. Then there exists a chain∅=B₀⊆B₁⊆ · · · ⊆B_k =Bwith#B_i =i, such that the subcodesC_i=hB_iiofCare indecomposable.

Proof. We define the setsB_i (and the codesC_i = hBii) iteratively byB₀ = ∅andB_i =B_i−1∪ {ci} for i ∈ {1, . . . , k}, wherec_i ∈ B\B_i−1is a codeword withsupp(c_i)∩supp(C_i−1) 6= ∅. Suitable codewordsc_ido indeed exist by the indecomposability ofC, as otherwiseC=C_i−1⊕ hB\B_i−1iwould be a non-trivial decomposition ofC fori 6= 1. We consider the simple graphG_i on the vertex setB_i where two distinct codewords are connected by an edge if their supports are not disjoint. By induction, all graphsG_iare connected.

LetEandE⁰be subcodes ofCiwithE+E⁰ =Ciandsupp(E)∩supp(E⁰) =∅. For any codeword c∈Cithere are codewordse∈Eande⁰ ∈E⁰withc=e+e⁰. Bysupp(e)∩supp(e⁰) =∅, we have w(c) =w(e) +w(e⁰). Ifcis of minimum weight, thene=0ore⁰ =0, so without restrictionc∈E. If c⁰ ∈Ciis another codeword of minimum weight withsupp(c)∩supp(c⁰)6=∅, then necessarilyc⁰ ∈E, too. The connectedness of the graphGishows that without restriction,b∈Efor allb∈Bi. Therefore, Ci =hBii=EandE⁰=0, implying thatCiis indecomposable.

We remark that the above proof still works as long asBconsists only of codewords of weight smaller than2d(C).

3.1. Non-binary codes. We start with the easier case of non-binary linear codes.

Theorem 2. Letq6= 2be a prime power and∆a positive integer. Letabe the largest integer such that q^adivides∆.

There area+ 1isomorphism types of indecomposable non-zero full-length∆-divisibleq-ary linear codes which are spanned by codewords of weight∆, one for each dimensionk ∈ {1, . . . , a+ 1}. It is given by the_qk−1^∆ -fold repetition of theq-ary simplex code of dimensionk.

Proof. By the discussion in Section 1, all stated codes are full-length,∆-divisible, indecomposable and spanned by codewords of weight∆. Furthermore, it is clear that a full-length ∆-divisible code of di- mension1 is equivalent to the∆-fold repetition of the simplex code of dimension1.² Now let C be an indecomposable full-length∆-divisible code of dimensionk ≥2 andB a basis ofC consisting of codewords of weight∆. By Lemma 3.2, there is ac ∈ B such that the codeD := hB \ {c}iis an indecomposable subcode ofCof codimension1. As a subcode,Dis∆-divisible, too. So by induction, D∼=0 ∆

q^k−2 ·Simq(k−1). Thus

# supp(D) = ∆

q^k−2· q^k−1−1 q−1 < q

q−1∆ =

1 + 1 q−1

∆.

By the indecomposability ofC,supp(c)∩supp(D)6=∅. Therefore, by Lemma 3.1 there is ac⁰∈B\{c}

with#(supp(c)∩supp(c⁰)) = ^q−1_q ∆. This implies that the lengthnofCis bounded by n <# supp(D) +1

q∆<

1 + 1

q−1+1 q

∆<2∆,

where we have usedq ≥ 3 in the last inequality. Now by the∆-divisibility, C is a code of constant weight∆. By Lemma 2.3,k≤a+ 1andCis equivalent to the code of the stated form.

3.2. Binary codes. We are going to use the same inductive approach in the binary case, too. For that purpose we prepare the following three lemmas, which investigate the extensions of repeated parity check codes, repeated simplex codes and repeated first order Reed-Muller codes.

Lemma 3.3. Letabe a positive integer and∆ = 2^a. LetCbe a binary indecomposable full-length∆- divisible code of dimensionk≥2with a basisBof codewords of weight∆. Letc∈B. IfhB\ {c}i ∼=0

∆

2 ·PC2(k−1), then (i) C∼=^∆₂ ·PC2(k)or

(ii) k= 3andC∼= ^∆₂ ·Sim₂(3)or (iii) k= 4andC∼= ^∆₂ ·RM2(4).

PROOF. LetC⁰=hB\ {c}i. Fork= 2, the codeC⁰is the span of a single codeword of weight∆, and by Lemma 3.1 and the indecomposability ofC, necessarilyC∼= ^∆₂ ·PC2(2).

So we may assumek≥3. We have# supp(C⁰) =k·^∆₂, and the positions insupp(C⁰)partition into ksetsI1, . . . , Ikof size#Is= ^∆₂ on which all codewords inC⁰agree. ForJ ⊆ {1, . . . , k}, letc^(J)be

2Note that the simplex code of length1is just the full Hamming spaceF¹q.

the characteristic vector of the setS

s∈JIs. The codewords ofC⁰are precisely the vectorsc^(J)with#J even.

Fors ∈ {1, . . . , k}, letbs = #(supp(c)∩Is). Assume0 < bt < ^∆₂ for somet∈ {1, . . . , k}. Let s ∈ {1, . . . , k}. The application of Lemma 3.1 to the codewordc^({s,t})shows thatbs+bt = ^∆₂ and in particular, 0 < bs < ^∆₂. Byk ≥ 3, there is as⁰ ∈ {1, . . . , k} \ {s, t}. Again, the application of Lemma 3.1 to the codewordsc^({s0,t})andc^({s,s0})shows thatb_s⁰ +b_t =b_s+b_s⁰ = ^∆₂. This implies bs=bs⁰ =bt= ^∆₄. So by varyingswe get thatbs = ^∆₄ for alls∈ {1, . . . , k}. Because ofP

sbs ≤ w(c) = ∆, we getk∈ {3,4}. In the casek= 3we see thatC∼=^∆₂ ·Sim2(3), and in the casek= 4, we check thatC∼=^∆₂ ·RM₂(4).

It remains to consider the casebs ∈ {0,^∆₂}for alls ∈ {1, . . . , k}. The number ofswithbs = ^∆₂ is at least1by the indecomposability ofCand at most2byPk

s=1b_s ≤ w(c). In the first case,C ∼=

∆

2 ·PC₂(k). The second case we havec∈C⁰, which is a contradiction.

Lemma 3.4. Letabe a non-negative integer, ∆ = 2^a andk ∈ {2, . . . , a+ 2}. Let C be a binary indecomposable full-length∆-divisible code of dimensionk. Assume thatC =hC⁰, ciwith a codeword cof weight∆andC⁰ ∼=0 ∆

2^k−2 ·Sim₂(k−1). Thena≥1and (i) k6=a+ 2andC∼= ₂k−1^∆ ·Sim₂(k)or

(ii) k6= 2andC∼=₂k−2^∆ ·RM2(k).

In particular,Cdoes not exist in the casea= 0.³

Proof. Because ofdim(C⁰) =k−1we havec /∈C⁰. By the indecomposability ofCthere is a codeword c⁰ ∈C⁰ withsupp(c)∩supp(c⁰) 6=∅. We havew(c) = ∆asC⁰ is a code of constant weight∆. By Lemma 3.1,#(supp(c)∩supp(c⁰)) = ^∆₂ and in particulara≥1. Therefore

n(C) = # supp(C)≤# supp(C⁰) +∆

2 = (2∆−2^a−k+2) +∆ 2 <5

2∆.

So by the∆-divisibility,C can only contain non-zero codewords of weight ∆and2∆. If there is no codeword of weight2∆, Lemma 2.3 givesk6=a+ 2andC∼=₂k−1^∆ ·Sim₂(k).

Otherwise, there is a codewordc⁰⁰ ∈Cof weight2∆. LetDbe the residual ofCwith respect toc⁰⁰. SinceCis full-length, so isD, and we getn(D) =n(C)−w(c⁰⁰)< ^∆₂. Furthermore, as the residual of a binary∆-divisible code,Dis^∆₂-divisible by Lemma 2.5, which implies thatDis a full-length zero code.

Thereforen(D) = 0and thusn(C) = 2∆. Soc⁰⁰is the all-one word andsupp(C⁰)⊆supp(c⁰⁰). From the simplex code structure ofC⁰we conclude thatC∼=₂k−2^∆ ·RM₂(k), and by the indecomposability of

C, we havek6= 2.

Lemma 3.5. Letabe a positive integer,∆ = 2^aandk∈ {4, . . . , a+ 3}. LetCbe a binary indecompos- able full-length∆-divisible code of dimensionkwith a basisBof codewords of weight∆. Letc∈B. If hB\ {c}i ∼=0 ∆

2^k−3 ·RM2(k−1), then

(i) a≥2,k≤a+ 2andC∼=₂k−2^∆ ·RM2(k)or (ii) k= 4andC∼=^∆₂ ·PC2(4).

Proof. LetC⁰ =hB\ {c}i. We have# supp(C⁰) = 2∆andC⁰contains a codewordeof weight2∆.

Lemma 3.1 and the indecomposability ofCyield2∆≤n(C)≤ ⁵₂∆. The residualDofCwith respect toeis a full-length^∆₂-divisible code of lengthn(D)≤ ⁵₂∆−2∆ =^∆₂. This leaves only the possibilities n(D)∈ {0,^∆₂}.

Case 1: n(D) = 0. Therefore, n(C) = 2∆. Soeis the all-one word in C and hence the unique codeword of weight2∆. Thus all but a single non-zero codeword are of weight∆. Therefore, C = hC⁰⁰, c⁰⁰i, whereC⁰⁰ is a subcode of dimensionk−1of constant weight ∆, andc⁰⁰ is a codeword of

3In the casea= 0, necessarilyk= 2, which is excluded in both stated cases.

weight∆. Lemma 2.3 shows thatk≤a+ 2andC⁰⁰∼=0 ∆

2^k−2 ·Sim2(k−1). Now by Lemma 3.4 and the fact thatCcontains a codeword of weight2∆, we haveC∼=₂k−2^∆ ·RM2(k).

Case 2:n(D) = ^∆₂. Therefore,n(C) = ⁵₂∆. The only non-zero weights ofCare∆and2∆. LetA∆

andA_2∆be their frequencies and let(B_i)be the weight distribution ofC^⊥. The first two Pless power moments give the linear equation system

1 1 1 2

A∆

A2∆

=

#C−1

5 4#C

with the unique solutionA∆= ³₄#C−2andA2∆= ¹₄#C+ 1. Now the third and the forth Pless power moment yield

B2= 4

#C∆²+3 8∆²−5

4∆ and B3=

2

#C−1 8

∆³.

The numberB3 must be non-negative, which is equivalent to#C ≤ 16 or k ≤ 4. So k = 4and C⁰ ∼=0 ∆

2 ·RM₂(3)∼= ^∆₂ ·PC₂(3). Now by Lemma 3.3 andn(C) = ⁵₂∆, we haveC∼= ^∆₂ ·PC₂(4).

We remark that the setting of the above lemma makes also sense fork= 3. We didn’t include it as the first order Reed-Muller-CodeRM2(2)∼=F²2(and its repetitions) are decomposable. Fork= 3, Case 1 is the same, and it is easily checked that there is no suitable codeCin Case 2.

Theorem 3. Let∆be a positive integer andathe largest integer with such that∆is divisible by2^a. The following codes form a complete and non-redundant list of all isomorphism types of binary indecompos- able full-length∆-divisible codes of dimensionk≥1that are spanned by codewords of weight∆.

(i) ₂k−1^∆ ·Sim2(k)with1≤k≤a+ 1, (ii) ₂k−2^∆ ·RM2(k)with3≤k≤a+ 2, (iii) ^∆₂ ·PC2(k)witha≥1andk≥4.

PROOF. All codes in the list have the stated properties, and by Remark 2.2, the list is non-redundant. By Lemma 2.4, it is enough to consider∆ = 2^a.

It is clear that the only sought-after code of dimension1is the code∆·Sim2(1). Now assume thatCis a binary indecomposable full-length∆-divisible code of dimensionk≥2having a basisBof codewords of weight∆. By Lemma 3.2, there is a codewordc∈Bsuch thatC⁰ :=hB\ciis indecomposable. By induction, we are in one of the following situations:

Case (i). 2 ≤ k ≤ a+ 2 andC⁰ ∼=0 ∆

2^k−2 ·Sim₂(k−1). By Lemma 3.4, either k 6= 2 and C∼=₂k−2^∆ ·RM₂(k), ork6=a+ 2andC∼= ₂k−1^∆ ·Sim₂(k).

Case (ii). 4 ≤k ≤a+ 3(so in particulara ≥1) andC⁰ ∼=0 ∆

2^k−3 ·RM2(k−1). By Lemma 3.5, eithera≥2andk≤a+ 2andC∼=₂k−2^∆ ·RM2(k), ork= 4andC∼= ^∆₂ ·PC2(k).

Case (iii).a≥1,k≥5andC⁰ ∼=0 ∆

2 ·PC2(k−1). By Lemma 3.3,C∼=^∆₂ ·PC2(k).

Our main result Theorem 1 is now a direct consequence of Theorem 2 and Theorem 3 and the discus- sion of decomposability of linear codes in Section 2.

We conclude this section by restating Theorem 1 for the important special cases of even, doubly- even and triply-even (i.e.2-divisible,4-divisible and8-divisible) binary codes. For the formulation of the statements, we use the isomorphisms2·Sim2(1)∼= PC2(1),Sim2(2)∼= PC2(2)andRM2(3)∼= PC2(3).

Corollary 3.6. LetCbe an even binary code that is spanned by codewords of weight2. ThenCisomor- phic to an essentially unique direct sum of binary parity check codes, possibly extended by zero positions.

The following doubly-even case is essentially [13, Th. 6.5].

Corollary 3.7. LetCbe a doubly-even binary code that is spanned by codewords of weight4. ThenCis isomorphic to an essentially unique direct sum of codes of the following form, possibly extended by zero positions.

(i) The binary simplex code of dimension3.

(ii) The binary first order Reed-Muller code of dimension4.

(iii) The2-fold repetition of a binary parity check code.

Corollary 3.8. LetCbe a triply-even binary code that is spanned by codewords of weight8. ThenCis isomorphic to an essentially unique direct sum of codes of the following form, possibly extended by zero positions.

(i) The2-fold repetition of the binary simplex code of dimension3.

(ii) The binary simplex code of dimension4.

(iii) The2-fold repetition of the binary first order Reed-Muller code of dimension4.

(iv) The binary first order Reed-Muller code of dimension5.

(v) The4-fold repetition of a binary parity check code.

4. AN APPLICATION

As an application, we consider binary projective∆-divisible codes of length4∆with∆ = 2^a. We will find an alternative proof of [10, Thm. 4] in the case of projective codes.

If a projective∆-divisible[4∆, k]2-codeCdoes not contain the all-one word1, we may look at the codeD = hC,1i, which is a projective∆-divisible[4∆, k+ 1]2−code. Therefore, we may restrict ourself to the codesDcontaining the all-one word, in the sense that all codesCwill show up as subcodes of these codesD.

Lemma 4.1. Let∆ = 2^aandCbe a binary projective∆-divisible[4∆, k]2-code containing the all-one word. LetC⁰be the subcode spanned by all codewords of weight∆. Then

(i) C⁰ ∼=0 ∆

2 ·PC₂(7)andk=a+ 6.

(ii) C⁰ ∼=0 ∆

2<sup>`−2</sup> ·RM<sub>2</sub>(`)⊕₂`−2<sup>∆</sup> ·RM<sub>2</sub>(`)andk=a+`+ 2where`∈ {2, . . . , a+ 2}.

(iii) C⁰ ∼=0 ∆

2 ·Sim2(2)⊕∆·Sim2(1)andk=a+ 4, (iv) C⁰ ∼=00andk=a+ 3.

PROOF. We haveA4∆ = 1. AsC is projective,B1 = B2 = 0. By the MacWilliams identities, we compute the weight enumerator ofCas

A₀= 1,

A∆= 2^k−a−1−4, A2∆= 2^k−2^k−a+ 6, A3∆= 2^k−a−1−4, A_4∆= 1.⁴

By Theorem 3,C⁰∼=0C1⊕. . .⊕Cs, where theCiare linear codes of the form (i) Ci= _2ki^∆₋₁·Sim2(ki)of dimensionki∈ {1, . . . , a+ 1}or

(ii) C_i= _2ki^∆

−2·RM₂(k_i)of dimensionk_i∈ {3, . . . , a+ 2}or (iii) Ci= ^∆₂ ·PC2(ki)of dimensionki≥4.

4It was clear before thatA∆=A3∆, asc7→c+1is a bijection between the sets of codewords of weight∆and3∆.

TABLE1. The decompositionsC⁰∼=0C₁⊕. . .⊕C_s

N^o C⁰∼=0 A∆ k condition

1 0 0 a+ 3

2 ^∆₂ ·PC2(7) 28 a+ 6

3 ^∆₂ ·PC2(6) 21 −

4 ^∆₂ ·PC₂(5) 15 −

5 ^∆₂ ·PC2(5)⊕∆·Sim2(1) 16 −

6 ^∆₂ ·PC₂(4) 10 −

7 ^∆₂ ·PC2(4)⊕∆·Sim2(1) 11 −

8 ^∆₂ ·PC2(4)⊕^∆₂ ·Sim2(2) 13 −

9 _2k^∆₁₋₂ ·RM₂(k₁) 2^k1−2 −

10 _2k^∆₁−2 ·RM₂(k₁)⊕_2k^∆₂−1 ·Sim₂(k₂) 2^k1+ 2^k2−3 −

11 ₂k^∆1−2 ·RM2(k1)⊕₂k^∆2−2 ·RM2(k2) 2^k1+ 2^k2−4 a+ 2 +k1 k1=k2

12 ₂k^∆1−2 ·RM2(k1)⊕∆·Sim2(1)⊕∆·Sim2(1) 2^k1 −

13 _2k^∆₁₋₁ ·Sim₂(k₁) 2^k1−1 −

14 _2k^∆₁−1 ·Sim₂(k₁)⊕₂k−1−1^∆ ·Sim₂(k₂) 2^k1+ 2^k2−2 a+ 4 {k₁, k₂}={1,2}

15 ₂k^∆1−1 ·Sim2(k1)⊕Sim2(1)⊕∆·Sim2(1) 2^k1+ 1 − 16 ∆·Sim₂(2)⊕∆·Sim₂(2)⊕∆·Sim₂(1) 7 − 17 ∆·Sim2(1)⊕∆·Sim2(1)⊕∆·Sim2(1)⊕∆·Sim2(1) 4 a+ 4

We haven_eff(C⁰)≤n_eff(C)andA_∆(C⁰) =A_∆(C), which gives the conditions neff(C1) +. . .+neff(Cs)≤4∆ and A∆(C1) +. . .+A∆(Cs) = 1

2∆#C−4.

The valuesA_∆andn_eff of the codesC_iare summarized below.

Ci neff(Ci) A∆(Ci)

∆

2ki−1 ·Sim2(ki) 2∆−_2ki^∆−1 2^ki−1

∆

2ki−2 ·RM2(ki) 2∆ 2^ki−2

∆

2 ·PC2(ki) ^∆₂ ·ki+ 1 ^ki₂⁺¹ Sorting the effective lengths of the possible codesC_i, we get

neff(∆·Sim2(1))

| {z }

=∆

< neff

∆

2 ·Sim2(2)

| {z }

=³₂∆

< . . . < neff(Sim2(a+ 1))

< neff

∆

2 ·RM2(3)

=. . .=neff(RM2(a+ 2))

| {z }

=2∆

< neff

∆

2 ·PC2(4)

| {z }

=⁵₂∆

< neff

∆

2 ·PC2(5)

| {z }

=3∆

< neff

∆

2 ·PC2(6)

| {z }

=⁷₂∆

< neff

∆

2 ·PC2(7)

| {z }

=4∆

.

Now we enumerate the possible codesC⁰ systematically by the restriction onn_eff(C_i). The result is shown in Table 1 From the expressionA_∆ = 2^k−a−1−4we see that A_∆+ 4 = 2^k−a−1 must be a power of2. In the case that this is possible, the resulting value ofkand possibly the required condition are displayed in corresponding columns. UsingSim₂(1)⊕Sim₂(1) = RM₂(2), lines 11 and 17 of the

table can be merged.

Theorem 4. Leta≥ 2be an integer,∆ = 2^a andCbe a binary projective∆-divisible[4∆, k]-code.

Thenk≤2a+ 4. In the case of maximum dimensionk= 2a+ 4, we have (i) C∼= RM2(a+ 2)⊕RM₂(a+ 2)or

(ii) a= 2andC∼=h2·PC2(7),(1111111100000000)i

PROOF. For the investigation of the maximum dimension, we may assume1∈C. LetC⁰be the subcode ofCspanned by the codewords of weight∆. Lemma 4.1 givesk≤2a+ 4. There are two cases where k= 2a+ 4is possible.

Case 1:C⁰∼= RM2(a+ 2)⊕RM2(a+ 2). Because ofdim(C⁰) = 2a+ 4we haveC=C⁰.

Case 2: a = 2andC⁰ ∼= 2·PC2(7). Heredim(C⁰) = 7anddim(C) = 8, soC = hC⁰, ciwith c /∈ C⁰. We haveA_2∆(C⁰) = ⁸₄

= 70and from the expression in the proof of Lemma 4.1 we have A_2∆(C) = 198, so we may assumew(c) = 8. To get a projective codeC, the support ofchas to contain exactly one member of each repeated pair of positions ofC⁰. We see that all codewords inC\C⁰are of weight8, soCis4-divisible. Moreover, the different choices ofclead to isomorphic codes.

Remark4.2. Theorem 4 is essentially [10, Thm. 4] in the case of projective codes.

The casea= 2is about4-divisible[16,8]₂-codes, which are binary Type II self-dual codes of length 16. These codes have first been classified in [14], where our two codes are calledA8⊕A8andE16. Up to isomorphism, these are the only Type II self-dual codes of length16.

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DEPARTMENT OFMATHEMATICS, UNIVERSITY OFBAYREUTH, BAYREUTH,GERMANY

Email address:michael.kiermaier@uni-bayreuth.de, sascha.kurz@uni-bayreuth.de