Classification of indecomposable 2^r-divisible codes spanned by by codewords of weight 2^r

CLASSIFICATION OF INDECOMPOSABLE2 -DIVISIBLE CODES SPANNED BY BY CODEWORDS OF WEIGHT2^r

SASCHA KURZ

ABSTRACT. We classify indecomposable binary linear codes whose weights of the codewords are divisible by2^rfor some integerrand that are spanned by the set of minimum weight codewords.

Keywords:linear codes, divisible codes, classification MSC:94B05.

1. INTRODUCTION

A binary[n, k]2 codeCis ak-dimensional subspace of then-dimensional vector spaceFⁿ2, i.e., we consider linear codes only. Elementsc ∈ C are called codewords and nis called the length of the code. The support of 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∆-divisible if the weight of all codewords is divisible by some positive integer∆≥1, see e.g. [8] for a survey. A classification of all∆-divisible codes seems out of reach unless the length is restricted to rather small values.

Given an[n, k]₂codeC, the[n, n−k]₂codeC^⊥=

x∈Fⁿ2. x^Ty= 0∀y∈C is called the orthog- onal, or dual ofC. A code is self-orthogonal ifC ⊆C^⊥ and self-dual ifC =C^⊥. A self-orthogonal code is2-divisible. In [6] self-orthogonal codes which are generated by codewords of weight4, which then are4-divisible, are completely characterized. Here we want to generalize that result, see [6, Theorem 6.5], and characterize2^r-divisible codes that are generated by codewords of weight2^r. Further related work includes the classical result of Bonisoli characterizing one-weight codes [1] and the generalization to two-weight codes where one of the weights is twice the other [3].

2. PRELIMINARIES

We call a code C non-trivial if its dimension dim(C) = k is at least 1. Using the abbreviation supp(C) = ∪_c∈Csupp(c), we call|supp(C)|the effective lengthneff ofC. Here we assume that all codes are non-trivial and that the effective lengthneff equals the lengthn(orn(C)to be more precise).

We emphasize this by speaking of an[n, k]₂code. A matrixGwith the property that the linear span of its rows generate the codeC, is a generator matrix ofC. A generator matrixGis called systematic if it starts with a unit matrix. Each code admits a systematic generator matrix. The assumption that the effective lengthn_eff is equal to the lengthnis equivalent to the property that generator matrices do not contain a zero-column. ByA_i(C)we denote the number of codewords of weightiinCand byB_i(C)the number of codewords of weightiinC^⊥. Mostly, we will just writeA_i andB_i, whenever the codeCis clear from the context. In our setting we haveA₀ =B₀ = 1andB₁ = 0. In general, theA_i and theB_iare related by the so-called MacWilliams identities, see e.g. [4]. The first four MacWilliams identities can be

1

rewritten to:

X

i>0

Ai = 2^k−1, (1)

X

i≥0

iAi = 2^k−1n, (2)

X

i≥0

i²Ai = 2^k−1(B2+n(n+ 1)/2), (3) X

i≥0

i³Ai = 2^k−2(3(B2n−B3) +n²(n+ 3)/2). (4) In this special form they are also called the first four(Pless) power moments, see [5]. The weight dis- tribution ofCis the sequenceA0, . . . , Anand the weight enumerator ofCis the polynomialw(C) = w(C;x) =Pn

i=0Aixⁱ.

Two codesC, C⁰ are equivalent, notated asC ' C⁰, if there exists a permutation in Sn sending Cinto C⁰. The direct sum of an[n, k]₂code Cand an[n⁰, k⁰]₂codeC⁰ is the [n+n⁰, k+k⁰]₂ code C⊕C⁰ ={(c1+c⁰₁, . . . , c_n+c⁰_n) : (c₁, . . . , c_n)∈C,(c⁰₁, . . . , c⁰_n)∈C⁰}. IfDcan be written asC⊕C⁰ it is called decomposable, otherwise indecomposable [7].

Lemma 2.1. LetC be an indecomposable[n, k]_q code. Ifk ≥2, thenCcontains an indecomposable ≤n−1, k−1

q codeC⁰as a subcode.

PROOF. Let Gbe a systematic generator matrix ofC. We will constructC⁰ by row-wise building up a generator matrix. To this end letRbe the set of rows and set C = ∅. For the start pick some row r∈ Radd it toCand remove it fromR. As long as# < k−1we choose some elementr ∈ Rwith supp(r)∩supp(c)6=∅for at least onec∈ C. SinceCis indecomposable such a rowrmust indeed exist.

Again, addrtoCand remove it fromR.

In other words, indecomposable codes can always be obtained by extending indecomposable subcodes.

Corollary 2.2. Each indecomposable[n, k]_q codeC contains a chainC0 ⊆C1 ⊆ · · · ⊆ Ck = Cof indecomposable subcodes such thatdim(C_i) =iand the effective length is strictly increasing.

Given some[n, k]₂codeCwe can restrict the coordinates of the codewords to some subsetI⊆N :=

{1, . . . , n}, i.e.,C_I = {cI : c∈C}, wherec_I denotes the codewordcrestricted to the positions inI.

Special cases are the codeC_supp(c)restricted to some codewordc ∈ Cand the corresponding residual codeC_N\supp(c). Note that the dimensions of both codes is at mostk−1but can be strictly less. IfC is2^rdivisible for some positive integerr, then a residual code ofCis2^r−1-divisible, see e.g. [9, Lemma 13], so that also the corresponding restricted code is2^r−1-divisible.

If all non-zero codewords of a binary linear code have the same weight, then the code is a replication of a simplex code, see [1]. For the reader’s convenience we prove a specialization of that result.

Lemma 2.3. LetCbe an[n, k]₂code where all non-zero codewords have weight2^a. Then,k ≤a+ 1 andC'S_k−1^a+1−k.

PROOF. By Lemma 3.1 there exists a codeC⁰ withC =C^0a+1−k. By construction all non-zero code- words ofC⁰have weight2^k−1. Using equations (1)-(3) we computen= 2^k−1andB₂= 0. Since there are only2^k−1different non-zero vectors inF^k2we haveC⁰'S_k−1⁰ , so thatC'S_k−1^a+1−k.

3. THE CHARACTERIZATION

We want to prove our main characterization result for indecomposable2^r-divisible[n, k]₂codes that are generated by codewords of weight2^r in Theorem 3.7. To this end, we describe some families of

CLASSIFICATION OF INDECOMPOSABLE2 -DIVISIBLE CODES SPANNED BY BY CODEWORDS OF WEIGHT2 3

codes and then derive some auxiliary results. So, bySlwe denote the(l+ 1)-dimensional simplex code, i.e., dim(S_l) = l+ 1andw_Sl(X) = 1 + (2^l+1−1)·X^2l, wherel ≥ 0. So, S_lis2^l-divisible and has effective lengthn = 2^l+1−1. ByA_l we denote the

2^l+1, l+ 2,2^l

1st-order Reed-Muller code, which geometrically corresponds to the affine(l+ 1)-flat, i.e.,S_l+1−S_l+in terms of point sets. So, dim(Al) =l+ 2andwA_l(X) = 1 + 2^l+2−2

·X^2l+ 1·X^2l+1, i.e., it is2^l-divisible and has effective lengthn = 2^l+1. ByR_lwe denote the l-dimensional code generate by the lcodewords having a1at position1and a second one at positioni+ 1for1≤i≤l. So,Rlhas dimensiondim(Rl) =l, effective lengthn =l+ 1and is 2¹-divisible. IfC is a code then byC^m we denote the code that arises if we replace every0by a block of2^mconsecutive zeroes and every1by a block of2^mconsecutive ones. So, especially we haveC⁰=C. In general the dimension does not change, the effective length is multiplied by2^mand a2^l-divisible code is turned into a2^l+m-divisible code. For the weight enumerator we have w(C^m;x) =w(C;x^m).

Lemma 3.1. Letq= p^ebe a prime power andCbe aq-ary linear code (considered as a powerset of Fⁿq) that isq^r-divisible, wherere∈N≥0. For each∅ ⊆M ⊆S⊆Cwith1≤ |S| ≤r+ 1we have that q^r+1−|S|divides#I_M,S(C), where

I_M,S(C) ={i∈supp(S) : i∈supp(c)∀c∈M ∧ i /∈supp(c)∀c∈S\M}.

PROOF. ForM = ∅we haveI_M,S(C) =∅, so that#I_M,S(C) = 0and the statement is trivially true.

In the following we assumeM 6=∅and prove by induction on#S. For the induction start letS ={c}.

Due to our assumption we haveM ={c}, so thatI_M,S(C) = # supp(c) = wt(c), which is divisible byq^r+1−|S| = q^r. Now let|S| ≥ 2 and¯c ∈ M be arbitrary. WithI = supp(¯c)we set C⁰ = C_I, i.e., the restricted code. As noted in Section 2,C⁰ isq^r−1-divisible (since|S| ≤ r+ 1impliesr ≥1).

We set M⁰ = {c_I : c∈M\{¯c}}andS⁰ = {c_I : c∈S\{¯c}}, so that ∅ ⊆ M⁰ ⊆ S⁰ ⊆ C⁰. Since

#S⁰= #S−1andIM,S(C) =IM⁰,S⁰(C⁰)the statement follows from the induction hypothesis.

Corollary 3.2. In the setting of Lemma 3.1 we have thatq^r+1−|S|divides the cardinality ofsupp(S).

PROOF. Since

supp(S) =∪_c∈Ssupp(c) = X

∅⊆M⊆S

IM,S(C),

the statement follows directly from Lemma 3.1.

Lemma 3.3. LetC =R^a_l for integersl≥1anda≥0,c⁰ be a further codeword with weight2^a+1and

∅ 6= supp(c⁰)∩supp(C)6= supp(C). IfC⁰:=hC, c⁰iis2^a+1-divisible, then eitherC⁰'R^a_l+1orl= 2, a≥1, andC⁰'S₂^a−1.

PROOF. As an abbreviation we set∆ := 2^a+1 and note that C is ∆-divisible. If l = 1, then C = {0, c}, wherewt(c) = ∆. From Lemma 3.1 we conclude that ^∆₂ divides|supp(C)∩supp(c⁰)|. Since supp(C) = supp(c)and∅ 6= supp(C)∩supp(c⁰)6= supp(C), we have|supp(C)∩supp(c⁰)| = ^∆₂. Thus,C⁰ 'R^a₂=R^a_l+1.

Now we assumel ≥ 2. For1 ≤ i ≤ l+ 1we setPi :=

j ∈N : ^∆₂(i−1) + 1≤j≤^∆₂i and f_i(c) := |supp(c)∩P_i|for each codewordc ∈ C⁰. Note thatf_i(c) ∈

0,^∆₂ for allc ∈ C and all 1 ≤ i ≤ l + 1. Moreover, for each 1 ≤ i < j ≤ l + 1 there exists a codeword c^i,j ∈ C with f_i(c^i,j) =f_j(c^i,j) = ^∆₂ andf_h(c^i,j) = 0otherwise. Now suppose that there is an index1≤i≤l+ 1 with0< fi(c⁰)<^∆₂. For each index1≤j≤l+ 1withi6=jwe have

wt(c^i,j+c⁰) = wt(c^i,j) + wt(c⁰)−2·wt(c^i,j∩c⁰) = 2∆−2fi(c⁰)−2fj(c⁰),

so thatwt(c^i,j+c⁰) = ∆andfi(c⁰) +fj(c⁰) = ^∆₂. Sincel ≥2there exists at least another index in {1, . . . , l+ 1} ∩ {i, j}, so that this impliesfh(c⁰) = ^∆₄ for all1 ≤h ≤l+ 1. Thus,∆ = wt(c⁰) >

Pl+1

h=1f_h(c⁰)impliesl = 2andC⁰ 'S₂^a−1. Otherwise we havef_h(c⁰)∈

0,^∆₂ for all1≤h≤l+ 1,

i.e., there exists an index1≤i≤l+ 1withfi(c⁰) = ^∆₂ andfh(c⁰) = 0otherwise. Ifi6= 1we consider

c⁰+c^1,ito conclude thatC⁰ =R^a_l+1.

Lemma 3.4. LetCbe a binary, non-trivial, indecomposable2¹-divisible linear code that is spanned by codewords of weight2. Then,C'R⁰_l for some integerl≥1.

PROOF. We will prove by induction on the dimensionk ofC. The induction startk = 1is obvious.

For the induction step letC⁰be an indecomposable subcode ofCwith dimensionk−1, see Lemma 2.1.

From the induction hypothesis we concludeC⁰ 'R⁰_k−1, so that Lemma 3.3 givesC'R_k⁰. Note thatS₀¹'R⁰₁,S₁⁰'R⁰₂, andA⁰₁'R⁰₃.

Lemma 3.5. LetCbe a binary, non-trivial, indecomposable∆-divisible linear code that is spanned by codewords of weight∆, where∆ = 2^a anda∈ N>0. Letc⁰ be a further codeword with weight∆and

∅ 6= supp(c⁰)∩supp(C)6= supp(C)such thatC⁰ :=hC, c⁰iis∆-divisible.

(1) IfC'S_a⁰thenC⁰'A⁰_a.

(2) IfC'S_a−1¹ thenC⁰'S_a⁰orC⁰ 'A¹_a−1. (3) Ifa≥1andC'A⁰_athena= 1andC⁰=R⁰₄. (4) Ifa≥2andC'A¹_a−1thena= 2andC⁰ 'R¹₄. (5) Ifa≥3andC'A²_a−2thena= 3andC⁰ 'R²₄.

PROOF. We note that1≤n(C⁰)−n(C)≤∆−1. Sincen(C)≤2∆in all cases the non-zero weights inC⁰are either∆or2∆.

(1) From equations (1)-(2) we computeA2∆= 2n(C⁰)−4∆ + 1, i.e.,A2∆≥1. LetDbe the residual code of a codeword of weight2∆inC⁰ (C⁰\C). By constructionDis ^∆₂-divisible, projective, and has an effective length of at most∆−2 <2·^∆₂ −1. Thus, Lemma 2.3 implies thatDis a trivial code, i.e.,n(D) = 0andn(C⁰) = 2∆. With this we haveA2∆= 1andC⁰'A⁰_a.

(2) From equations (1)-(2) we computeA∆ = 4∆−2−n(C⁰)andA2∆ = n(C⁰)−2∆ + 1, i.e., n(C⁰)≥2∆−1. Ifn(C⁰) = 2∆−1thenA2∆= 0and Lemma 2.3 givesC⁰'S_a⁰. Ifn(C⁰) = 2∆

thenA_2∆= 1and adding the all-one word toCgivesC⁰ 'A¹_a−1. In the remaining cases we have n(C⁰)>2∆andA_2∆≥1. LetDbe the residual code of a codeword of weight2∆inC⁰(C⁰\C).

By constructionDis ^∆₂-divisible, has column multiplicity at most2, and has an effective length of at most∆−3<2·^∆₂ −2. Thus, Lemma 2.3 implies thatDis a trivial code – contradiction. (The two possibilities with column multiplicity1or2would have an effective length of∆−1or∆−2, respectively.)

(3) From equations (1)-(2) we computeA_∆= 16∆−2−4n(C⁰)andA_2∆= 4n(C⁰)−8∆+1. LetDbe the residual code of a codeword of weight2∆inC⁰\C. By constructionDis^∆₂-divisible, projective, contains the all-1codeword, and has an effective length of at most∆−1. Thus, Lemma 2.3 implies thatD'S₀^a−1, wherea= 1. So,C=R⁰₃and Lemma 3.3 yieldsC⁰=R⁰₄.

(4) From equations (1)-(2) we computeA_∆ = 8∆−2−2n(C⁰)andA_2∆ = 2n(C⁰)−4∆ + 1. Let Dbe the residual code of a codeword of weight2∆inC⁰\C. By constructionDis ^∆₂-divisible, has maximum column multiplicity at most2, contains the all-1codeword, and has an effective length of at most∆−1. Thus, Lemma 2.3 implies that eitherD 'S⁰₀ orD ' S₀¹. In the first case we have∆ = 2 anda = 1, which is not possible. In the second case we have∆ = 4,a = 2, and C'A¹₁'R¹₃, so that Lemma 3.3 impliesC⁰ 'R¹₄.

(5) From equations (1)-(2) we computeA_∆ = 4∆−2−n(C⁰)andA_2∆=n(C⁰)−2∆ + 1. LetD be the residual code of a codeword of weight2∆inC⁰\C. By constructionDis ^∆₂-divisible, has maximum column multiplicity at most4, contains the all-1codeword, and has an effective length of at most∆−1. Thus, Lemma 2.3 implies that eitherD'S₀⁰,D'S₀¹, orD'S₀². Since we assume a≥3, onlya= 3and∆ = 8is possible, whereC'R²₃, so that Lemma 3.3 impliesC⁰'R₄².

CLASSIFICATION OF INDECOMPOSABLE2 -DIVISIBLE CODES SPANNED BY BY CODEWORDS OF WEIGHT2 5

Note that if we drop the conditionsupp(C⁰) 6= supp(C), thenA¹_a−1can be extended toA⁰_aandA²_a−2 can be extended toA¹_a−1.

Lemma 3.6. LetCbe a binary, non-trivial, indecomposable2²-divisible linear code that is spanned by codewords of weight4. Then,C'R¹_l for some integerl≥1or eitherC'S_2−l^l orC'A^l_2−lfor some l∈ {0,1}.

PROOF. First note that the mentioned families of codes satisfy all assumptions. Ifdim(C) ≤ 2then Lemma 3.1 implies that there is some codeC⁰withC=C⁰¹, i.e., we can apply Lemma 3.4. Ifdim(C)≥ 3 we apply Corollary 2.2 and consider the corresponding chainC0 ( C1 ( · · · ( Ck = C, where k= dim(C). Lemma 3.1 gives the existence of a binary, non-trivial, indecomposable2¹-divisible linear codeC⁰ withC2 =C⁰²that is spanned by codewords of weight2. Thus, Lemma 3.4 givesC⁰ ' R⁰₂ andC2 ' R¹₂. Lemma 3.3 then givesC3 ' R¹₃ orC3 ' S₂⁰. IfC3 ' R¹₃then recursively applying Lemma 3.3 yieldsC_l 'D¹_l for all3≤l ≤k. IfC₃ 'S₂⁰andk≥4, then Lemma 3.5 givesC₄ 'A⁰₂

andk= 4(sinceA⁰₂cannot be extended).

Note thatS₁¹'R¹₂andA¹₁'R¹₃.

Theorem 3.7. For a positive integeraletCbe a binary, non-trivial, indecomposable2^a-divisible linear code that is spanned by codewords of weight2^a. Then,C ' R^a−1_l for some integerl ≥ 1 or either C'S_a−l^l orC'A^l_a−lfor somel∈ {0,1, . . . , a−1}.

PROOF. We prove by induction ona. Lemma 3.4 and Lemma 3.6 give the induction start, so that we can assumea≥3in the following. First note that the mentioned families of codes satisfy all assumptions. If dim(C)≤athen Lemma 3.1 implies that there is some codeC⁰ withC =C⁰¹, i.e., we can apply the induction hypothesis. Ifdim(C)≥a+ 1we apply Corollary 2.2 and consider the corresponding chain C0(C1(· · ·(Ck =C, wherek= dim(C). Lemma 3.1 gives the existence of a binary, non-trivial, indecomposable2^a−1-divisible linear codeC⁰ withCa =C⁰² that is spanned by codewords of weight 2^a−1. Then the induction hypothesis gives that eitherCa 'R^a−1_a ,Ca 'S_a−1¹ , orCa 'A²_a−2. In the first case recursively applying Lemma 3.3 yieldsC_l'R^a−1_l for alla≤l ≤k. If eitherC_a 'S_a−1¹ or Ca 'A²_a−2we can apply Lemma 3.5 to concludeCa+1 'S_a⁰,Ca+1 'A¹_a−1, ora= 3andC4 'R²₄. In the latter case we haveCl 'R_l²for all4≤l ≤kdue to Lemma 3.3. Otherwise eitherk=a+ 1or

C_a+2'A⁰_aandk=a+ 2due to Lemma 3.5.

4. AN APPLICATION TO PROJECTIVE3-WEIGHT CODES

When deciding the question whether a code with certain parameters exist one often checks whether the MacWilliams identities admit a non-negative integer solution. If so, then sometimes more combinatorial are necessary. In the proof of e.g. [2, Lemma 24] the existence of an[51,9]₂code with weight enumerator w(C) = 1 + 2x⁸+ 406x²⁴+ 103x³²had to be excluded in a subcase. Since the sum of two codewords of weight8would have a weight between8and16this is impossible. Using the classification result of Theorem 3.7 this can easily be generalized.

Proposition 4.1. LetCbe a∆-divisible[n, k]₂ code, where∆ = 2^rfor some positive integerr. IfC does not contain a codeword of weight2∆, thenA∆∈

2ⁱ−1 : 0≤i≤r+ 1 .

PROOF. LetC⁰ be the subcode ofCspanned by the codewords of weight∆andC⁰ =C1⊕ · · · ⊕Cl

the up to permutation unique decomposition into indecomposable codes. SinceC⁰ does not contain a codeword of weight2∆we havel≤1. Forl= 0we obviously haveA_∆= 0. Ifl= 1, then Theorem 3.7

givesC1'S_i^r−i, where0≤i≤r, andA∆= 2ⁱ⁺¹−1.

In general, if we know that an[n, k]₂code is∆ := 2^r-divisible and contains some codewords of weight

∆one can consider the decompositionC⁰=C₁⊕ · · · ⊕C_lof the subcodeC⁰spanned by codewords of weight∆. Obviously, we have

(1) w(C⁰) =Ql

i=1w(Ci), i.e., especiallyA∆(C⁰) =Pl

i=1A∆(Ci);

(2) dim(C)≥dim(C⁰) =Pl

i=1dim(Ci);

(3) n(C)≥n(C⁰) =Pl

i=1n(Ci);

(4) ω(C)≥ω(C⁰) =Pl

i=1ω(C_i), whereω(D)denotes the maximum weight of a codeword inD.

With respect to Theorem 3.7 we remark

(1) A_∆(S^l_r−l) = 2^r+1−l−1,dim(S_r−l^l ) = r+ 1−l,n(S_r−l^l ) = 2^r+1−2^l, andω(S_r−l^l ) = ∆for 0≤l≤r;

(2) A∆(A^l_r−l) = 2^r+2−l−2,dim(A^l_r−l) =r+ 2−l,n(A^l_r−l) = 2∆ = 2^r+1, andω(A^l_r−l) = 2∆for 0≤l≤r−1;

(3) A∆(R^r−1_l ) = ^l+1₂

,dim(R^r−1_l ) =l,n(R^r−1_l ) =^∆₂ ·(l+ 1), andω(R^r−1_l ) =dl/2e ·∆forl≥1.

A more sophisticated example, compared to Proposition 4.1, occurs in the area of binary projective3- weight codes. Projective codes, i.e., those withB2= 0, having few weights have a lot of applications and have been studied widely in the literature. Here we consider[n, k]₂codes with weights in{0,∆,2∆,3∆}

and lengthn= 4∆, where∆ = 2^rfor some positive integerr.

Theorem 4.2. For an integerr ≥ 2let∆ = 2^randCbe a projective∆-divisible[4∆, k]₂code with non-zero weights in{∆,2∆,3∆}. Thenk≤2r+ 3. Ifk= 2r+ 3andr≥3thenCis isomorphic to a code with generator matrix





A⁰_r−1 A⁰_r−1 0 0 0 0 S_r⁰ 0

1 0 1 1



,

where0and1are matrices of approbriate sizes that entirely consist of 0’s or 1’s, respectively

PROOF. Using equations (1)-(3) and B2 = 0 we computeA∆ = 2^k−r−1−3 ≥ 1. Consider the decomposition C⁰ = C₁⊕ · · · ⊕C_l of the subcode C⁰ spanned by codewords of weight ∆. Since ω(C) = 3∆, we have 1 ≤ l ≤ 3. If ω(C_i) = ∆ for all 1 ≤ i ≤ l, i.e., C_i = S_r−j^ji

i for some 0≤ji≤r−1, thenA∆(C⁰) =Pl

i=1A∆(Ci)≤l·(2∆−1)≤3· 2^r+1−1

, so thatk <2r+ 4. If ω(C1) = 2∆, then due to Theorem 3.7 we have eitherC1'R^r−1₃ ,C1'R^r−1₄ , orC1'A^j_r−jfor some 0≤j ≤r−1, so thatA_∆(C₁)≤2^r+2−2. Since thenl ≤2,ω(C₂)≤∆, andA_∆(C₂)≤2^r+1−1, we haveA_∆(C⁰) =Pl

i=1A_∆(C_i)≤3· 2^r+1−1

, so thatk <2r+ 4. Ifω(C₁)≥3∆, thenl= 1and ω(C₁) = 3∆, so that Theorem 3.7 givesC₁'R^r−1₅ orC₁'R^r−1₆ , i.e.,A_∆(C⁰)≤21≤3· 2^r+1−1

, so thatk <2r+ 4. Thus, we havek≤2r+ 3in all cases.

Fork= 2r+ 3we need a more detailed analysis of the possible decompositionsC⁰ =C₁⊕ · · · ⊕C_l. First we noteω(Ci)∈ {∆,2∆,3∆},A∆ = 2^r+2−3 ≥ 1, so thatCi 6' A⁰_r, and1 ≤l ≤3. Let us start to consider the caseω(C_i) = ∆for alli, i.e.,A_∆= 2^r+1−ji−1for some0≤j_i≤r(C_i=S_r−j^ji

i

for some0 ≤j_i ≤r). Ifj_i ≥1for alli, thenA_∆(C⁰)≤3·(2^r−1) <2^r+2−3, so that we assume j₁ = 0. Since2^r+2−3 = 2^r+1−1is equivalent tor= 0, we havel ≥2. Ifl = 2andj₂ = 0, then A_∆(C⁰)≥2^r+2−2>2^r+2−3. Ifl= 2andj₂ ≤1, thenA_∆(C⁰)≤2^r+1−1 + 2^r−1<2^r+2−3 forr ≥ 1. Thus, we havel = 3. Ifj2 = 0 orj3 = 0, thenA∆(C⁰) ≥ 2· 2^r+1−1

>2^r+2−3.

Ifj2 ≥ 1,j3 ≥ 1, andj2+j3 ≥ 3, thenA∆(C⁰) ≤ 2^r+1−1 + 2^r−1 + 2^r−1−1 < 2^r+2 −3.

The only possibility withA∆(C⁰) = 2^r+2−3isj1 = 0,j2 =j3 = 1. However, in this case we have n(C⁰) = 2^r+1−1

+ 2^r+1−2

+ 2^r+1−2

= 2^r+2+ 2^r+1−5

>2^r+2=nforr≥2.

Ifω(Ci) = 3 for somei, thenl = 3and Theorem 3.7 givesC1 ' R^r−1₅ or C1 ' R^r−1₆ , so that A∆(C⁰) = ⁶₂

= 15orA∆(C⁰) = ⁷₂

= 21. Since2^r+2−3 <15forr ≤2and2^r+2−3>21for r≤3, this is not possible. Thus, there exists an indexiwithω(Ci) = 2. W.l.o.g. we assumeω(C1) = 2.

From Theorem 3.7 we concludeC1'R^r−1₄ orC1'A^j_r−jfor some integer0≤j ≤r−1. Ifl= 2, then ω(C2) = ∆, so that in any case we haveA∆(C⁰) =A∆(C1) + 2^x−1for some integer0≤x≤r+ 1.

IfC₁ ' R^r−1₄ , then the equationA_∆(C⁰) = 2^r+2−3 = 10 + 2^x−1has the unique integer solution

CLASSIFICATION OF INDECOMPOSABLE2 -DIVISIBLE CODES SPANNED BY BY CODEWORDS OF WEIGHT2 7

r = 2andx= 2, which corresponds toC⁰ ' R¹₄⊕S¹₁ ' R¹₄⊕R¹₂. (The equation is equivalent to 2^r+2 = 12 + 2^x, so thatr≥2. Forr≥2we havex≥5, so that the left hand side is divisible by8while the right hand side is not.) In the remaining cases we haveC1'A^j_r−j, so thatA∆(C1) = 2^r+2−j−2.

Thus, we have to consider the Diophantine equationA∆(C⁰) = 2^r+2−3 = 2^y−2 + 2^x−1, where y=r+ 2−j. The only integral solution isy=x=r+ 1, i.e.,j= 1,C1'A¹_r−1, andC2=S_r⁰.

To sum up, fork= 2r+3andr≥2, up to permutations, the only possibility isl= 2,C₁'A¹_r−1, and C2=S⁰_rwithdim(C⁰) = 2r+ 2andn(C⁰) = 2^r+2−1 = 4∆−1. Having fixedk= 2r+ 3we can use equations (1)-(3) to computeA∆(C) = 2^r+2−3andA3∆(C) = 2^r+2−1. Sincedim(C)−dim(C⁰) = 1 andA3∆(C⁰) = 2^r+1−1< 2^r+2−1, we can assume thatC =hC⁰, c⁰iwithwt(c⁰) = 3∆. SinceC is projective from the2∆coordinates of theC1 'A¹_r−1-part exactly the half have to be ones (and the other half have to be zeroes) inc⁰. Thus,c⁰has a one in each of the remaining2∆coordinates, so thatC is isomorphic to a code with generator matrix

G=





A⁰_r−1 A⁰_r−1 0 0 0 0 S_r⁰ 0

1 0 1 1



,

We remark that forr= 1there exists a corresponding code of dimension2r+ 4, i.e., there is a unique projective[8,6]₂code with weight enumerator1 + 13x²+ 35⁴+ 15x⁶. Forr= 2there exist more than one isomorphism types of codes of dimension2r+ 3, i.e., there exist exactly two isomorphism types of projective[16,7]₂codes with weight enumerator1 + 13x⁴+ 99x⁸+ 14x¹². (For the additional code we haveC⁰=R¹₄⊕R¹₂,dim(C⁰) = 6, andn(C⁰) = 16. Sincen(C) =n(C⁰),dim(C)−dim(C⁰) = 1, and Cis projective, we haveC=C⁰².) Forr= 3the non-existence of a projective[32,10]₂code with weight enumerator1 + 61x⁸+ 899x¹⁶+ 63x²⁴can not be concluded directly from the MacWilliam identities.

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SASCHAKURZ, UNIVERSITY OFBAYREUTH, 95440 BAYREUTH, GERMANY Email address:sascha.kurz@uni-bayreuth.de