On the Automorphism Groups of the Z2Z4-Linear 1-Perfect and Preparata-Like Codes

0.15 MB·

by  Denis Krotov

Designs, Codes and Cryptography manuscript No.
(will be inserted by the editor)
On the Automorphism Groups of the Z Z -Linear
2 4
1-Perfect and Preparata-Like Codes
Denis S. Krotov
Received: 2015-07-07/Accepted: ????-??-??
6
1
0 Abstract We consider the symmetry group of a Z Z -linear code with pa-
2 4
2
rameters of a 1-perfect, extended 1-perfect, or Preparata-like code. We show

n that, provided the code length is greater than 16, this group consists only
a
of symmetries that preserve the Z Z structure. We find the orders of the
J 2 4
9 symmetry groups of the Z2Z4-linear (extended) 1-perfect codes.
2 Keywords additive codes · Z Z -linear codes · 1-perfect codes · Preparata-
2 4

like codes · automorphism group · symmetry group
]
T
Mathematics Subject Classification (2010) MSC 94B25
I
.
s
c
[ 1 Introduction

1
v Spaces considered in coding theory usually have both metrical and algebraic
6 structures. From the point of view of the parameters of an error-correcting
3
code, the metrical one is the most important, while the algebraic properties
0
give an advantage in constructing codes, in developing coding and decoding
0
algorithms, or in different applications. In some cases, there are some “rigid”
0
. connections between metrical and algebraic structures. For example, if the q-
2
ary Hamming metric space with q = pm = 21,31,22 is considered as a vector
0
6 space over the field GF(p), then any isometry of the space is necessarily an
1 affinetransformation.Thisisnotthecaseforanyprimepowerq ≥5.However,
: the stabilizer of some codes in the group of space isometries consists of affine
v
i transformationsonly.So,informally,fromthe point ofview ofsuchcodes,the
X
algebraic structure is rigidly connected with the metrical one. For example,
r
a
Theresultswerepresentedinpartatthe3thInternationalCastleMeetingonCodingTheory
andApplicationsin2011.
D.Krotov
SobolevInstituteofMathematics,pr.AkademikaKoptyuga 4,Novosibirsk630090, Russia
E-mail:[email protected]
2 DenisS.Krotov
this was provedfor the Hamming codes for an arbitrary q [12]. In the current
paper, we prove a similar result for the Z Z -linear perfect, extended perfect,
2 4
andPreparata-likecodes withrespectto a Z Z algebraicstructure,whichis,
2 4
after the field structure, one of most important in coding theory. In contrast,
the situation with the Z Z -linear Hadamard codes is different [17]: the au-
2 4
tomorphism group of such a code is larger than the group of automorphisms
preserving the Z Z -linear structure.
2 4
In Sections 2–6, we give basic definitions and facts about the concepts
discussedinthepaper.Themainresultofthepaperandimportantcorollaries
are formulated in Section 7. Section 8 contains a proof of the main result,
whichstatesthatacodefromtheconsideredclassescanadmitonlyoneZ Z -
2 4
linear structure. As a direction for further research,it would be interesting to
generalize this result to a more wide class of Z Z -linear codes. The study of
2 4
automorphism groups is motivated by their role in decoding algorithms, see
e.g. [3].
2 Z2Z4-Linear Codes
A binary code C ⊂{0,1}n is called Z Z -linear if for some order-2 permuta-
2 4
tion (involution) π of the set {0,1,...,n−1} of coordinates, C is closed with
respect to the operation ∗ , where
π
def
x∗ y = x+y+(x+π(x))·(y+π(y)),
π
+and·beingthecoordinatewisemodulo-2additionandmultiplicationrespec-
tively (here andelsewhere,the actionof a permutationσ :{0,1,...,n−1}→
{0,1,...,n − 1} on x = (x0,x1,...,xn−1) ∈ {0,1}n is defined as σ(x) d=ef
(xσ−1(0),xσ−1(1),...,xσ−1(n−1))). Given an involution π, we will say that co-
ordinates i and j are adjacent if π(i) = j; if π(i) = i, then i is self-adjacent.
Clearly, the value of the result z = x∗ y in some coordinate i depends only
π
onthe values ofxandy inthe ithandπ(i)th coordinates.So,consideringthe
resultof∗ intwodifferentadjacentcoordinates,wecandetermine the values
π
by Table 1(a), while the case π(i)=i corresponds to Table 1(b).
Tables 1(a) and 1(b) are the value tables of groups isomorphic to Z and
4
Z ,respectively.Consequently,({0,1}n,∗ )isisomorphictothe groupZαZβ,
2 π 2 4
where α is the number of self-adjacentcoordinates and β =(n−α)/2. In this
Table 1 Valuesof∗π fortwodifferentadjacentcoordinates andforaself-adjacentcoordi-
nate
00 01 11 10
00 00 01 11 10 01
(a) 01 011110 00 (b) 001
11 111000 01 110
10 10 00 01 11
OntheAutomorphismGroupsofZ2Z4-LinearCodes 3
case we will say that π is a Z Z structure of type (α,β). The binary codes
2 4
closed with respect to ∗ are known as Z Z -linear codes of type (α,β). The
π 2 4
Z Z -linear codes with α=n are called linear; with α=0, Z -linear.
2 4 4
3 Z2Z4-Additive Codes, Gray Map, Duality
In this section, we consider an alternative way to define Z Z -linear codes
2 4
and related concepts. In the literature, this way is more popular than the
definition given in Section 2; however, for presenting results of the current
paper the last one is more convenient. The content of the section is not used
in the formulation of the main result of the paper (Theorem 1) and its proof,
but the concepts defined here are exploited in the proof of Corollary 2 (to
derive the order of a Z Z -linear (extended) 1-perfect code from Theorem 1
2 4
and known facts) and in the formulation of Corollary 3.
A code C ⊆ {0,1}α×{0,1,2,3}β in the mixed Z –Z alphabet is called
2 4
additive (Z Z -additive)ifitisclosedwithrespecttothecoordinatewiseaddi-
2 4
tion,modulo2inthefirstαcoordinatesandmodulo4inthelastβcoordinates.
The one to one correspondence Φ:{0,1}α×{0,1,2,3}β →{0,1}α+2β known
as the Gray map is defined as follows:
Φ((x0,...,xα−1,y0,...,yβ−1))=(x0,...,xα−1,φ(y0),...,φ(yβ−1)),
where φ(0) = (0,0), φ(1) = (0,1), φ(2) = (1,1), φ(3) = (1,0). The following
straightforwardfactmeansthataZ Z -linearcodecanbedefinedastheimage
2 4
of a Z Z -additive code under the Gray map and a coordinate permutation.
2 4
Proposition 1 A code C ⊆{0,1}α×{0,1,2,3}β is additive if and only if its
image Φ(C) under the Gray map is closed under the operation ∗ , where
π
π =(α α+1)(α+2 α+3)...(α+2β−2 α+2β−1). (1)
′ ′
The inner product [x,y]of twowordsx=(x0,...,xα−1,x0,...,xβ−1)and
y =(y0,...,yα−1,y0′,...,yβ′−1) from {0,1}α×{0,1,2,3}β is defined as
def ′ ′ ′ ′
[x,y] = 2x0y0+...+2xα−1yα−1+x0y0+...+xβ−1yβ−1 mod4. (2)
For a Z Z -additive code C ⊆{0,1}α×{0,1,2,3}β, its dual C⊥ is defined as
2 4
C⊥ d=ef {x∈{0,1}α×{0,1,2,3}β |[x,y]=0 for all y ∈C}. (3)
⊥ ⊥ ⊥
Readily, C is also an additive code. Moreover,(C ) =C, see, e.g., [5].
4 DenisS.Krotov
4 Symmetry Group
Let S be the set of permutations of {0,1,...,n−1}.The symmetry group of
n
a code C ⊂{0,1}n is defined as
def
Sym(C) = {σ ∈S |σ(x)∈C for all x∈C}.
n
GivenaZ Z -structureπ,wewillalsoconsiderthegroup of Z Z -symmetries
2 4 2 4
Sym (C) as a subgroup of Sym(C) consisting of symmetries σ that commute
π
with the involution π:
def
Sym (C) = {σ ∈Sym(C)|σ(π(i))=π(σ(i)) for all i∈{0,1,...,n−1}}.
π
In other words, Sym (C) is the intersection of Sym(C) with the automor-
π
phism group of the group ({0,1}n,∗ ) (which is, by definition, the set of all
π
permutations σ of {0,1}n such that σ(x)∗ σ(y)=σ(x∗ y) for every x, y).
π π
ThegroupSym (C)hasanaturaltreatmentintermsofthepreimageofC
π
under the Gray map. Indeed, if C =Φ(C) for some C ⊆{0,1}α×{0,1,2,3}β
and π is of form (1), then
Sym (C)={ΦσΦ−1 |σ ∈MAut(C)}, (4)
π
whereMAut(C),the monomial automorphism group ofC,isthe stabilizerofC
in the groupof monomialtransformationsof{0,1}α×{0,1,2,3}β (recallthat
a monomial transformation consists of a coordinate permutation followed by
sign changes in some quaternary coordinates).
To prove one of the corollaries from the main theorem, we will need the
following simple known fact.
⊥
Proposition 2 ForeveryZ Z -additivecodeC,itholdsMAut(C)=MAut(C ).
2 4
Proof At first, we see from(2) that [σ−1(x),y]=[x,σ(y)] for every monomial
⊥ ⊥
transformation σ. This identity can be utilised to derive σ(C ) = (σ(C))
from (3):
σ(C⊥)= {σ(x)|[x,y]=0 ∀y ∈C}={x′ |[σ−1(x′),y]=0 ∀y ∈C}
′ ′ ′ ′ ′ ′ ⊥
= {x |[x,σ(y)]=0 ∀y ∈C}={x |[x,y ]=0 ∀y ∈σ(C)}=(σ(C)) .
⊥ ⊥
If σ ∈ MAut(C), then σ(C) = C and hence σ(C ) = C . We conclude that
⊥ ⊥ ⊥
MAut(C)⊆MAut(C ). From (C ) =C, we get the inverse inclusion.
It is worth to mention the full automorphism group Aut(C) of a binary
code C, which is the stabilizer of the code in the group of isometries of the
Hamming space. We only observe that for a Z Z -linear code C, this group
2 4
is the product of Sym(C) and the group of translations {tr | c ∈ C}, where
c
def
tr (x) = c∗ x. As follows, |Aut(C)|=|C|·|Sym(C)|.
c π
OntheAutomorphismGroupsofZ2Z4-LinearCodes 5
5 Perfect and Extended Perfect Codes
A binary code C ⊂ {0,1}n is called 1-perfect (extended 1-perfect) if its car-
dinality is 2n/(n+1) (respectively, 2n−1/n) and the distance between every
two distinct codewords is at least 3 (respectively, 4), where the (Hamming)
distance is defined as the number of positions in which the words differ. Note
that the denominator (n+1) (respectively, n) coincides with the cardinality
of a radius-1 ball (respectively, sphere)and must be a power of 2 for the exis-
tenceofcorrespondingcodes.So,acharacterizingpropertyofa1-perfectcode
is that everybinary wordis at distance at most 1 from exactly one codeword.
The codewords of an extended 1-perfect code have the same parity (i.e., the
parity of the weight, the number of ones in the word), and every word of the
other parity is at distance 1 from exactly one codeword.
There is a characterization of Z Z -linear 1-perfect and Z Z -linear ex-
2 4 2 4
tended 1-perfect binary codes, see [8] (Z Z -linear 1-perfect codes), [15,16]
2 4
(Z -linear extended 1-perfect codes), and [6] (complete description). Recall
4
that the rank of a binary code is the dimension of its linear closure over Z .
2
Proposition 3 ([8,15,6]) (a) For any r and t ≥ 4 such that t/2 ≤ r ≤ t,
there is exactly one Z Z -linear 1-perfect code (extended 1-perfect code) of
2 4
type (2r − 1,2t−1 −2r−1) (respectively, (2r,2t−1 −2r−1)), up to coordinate
permutation.
(b) For any t ≥ 4, there are exactly ⌊(t+1)/2⌋ Z Z -linear extended 1-
2 4
perfect codes of type (0,2t−1) (i.e., Z -linear), up to coordinate permutation;
4
all these codes have different ranks 2t−r−1, r =⌊t/2⌋,...,t−1, except for
the case t=4, r =3, where the corresponding code is linear.
(c) All codes from (a) and (b) are pairwise nonequivalent. There are no
other Z Z -linear 1-perfect codes or Z Z -linear extended 1-perfect codes.
2 4 2 4
6 Preparata-Like Codes
A binary code C ⊂ {0,1}n is called Preparata-like if its cardinality is 2n/n2
andthedistancebetweeneverytwodistinctcodewordsisatleast6.Suchcodes
exist if and only if n is a power of 4 [20]. The original Preparata code [20],
and the generalizations [11], [1], [10] are not Z -linear if n > 16. A class of
4
Z -linearPreparata-likecodeswasconstructedin[13]foreveryn=2t+1 ≥16,
4
t odd; codes nonequivalent to that from [13] were found in [9]. As was shown
in [14, Theorem 5.11], there are many nonequivalent Z -linear Preparata-like
4
codes of the same length (their number grows faster than any polynomial in
n; however, there are some restrictions on n = 2t+1: t is not a prime nor the
product of two primes). We will use the following two facts, which make our
results concerning Preparata-like codes simple corollaries from the results on
extended 1-perfect codes.
Proposition 4 ([21]) For every Preparata-like code P, there exists a unique
extended 1-perfect code C including P.
6 DenisS.Krotov
Proposition 5 ([7]) Assume that a Preparata-like code P is closed with re-
spect to the operation ∗ , where π is a Z Z structure. Then the extended
π 2 4
1-perfect code C including P is also closed with respect to ∗ .
π
In[7],itwasshownthatanyZ Z -linearPreparata-likecodeisnecessarily
2 4
Z -linear, i.e., the involution π has no fixed points.
4
Remark 1 Inthecurrentwork,weconsiderthedistance-6Preparata-likecodes,
sometimesreferredtoastheextendedPreparata-likecodes.Inone-to-onecor-
respondence with such codes are the distance-5 Preparata-like codes, some-
timescalledthepuncturedPreparata-likecodes(infact,theoriginalPreparata
codes[20]werepresentedinterms ofdistance-5codes).ReformulatingPropo-
sition4,everypunctured Preparata-likecodeis includedina unique 1-perfect
code.Formally,wecanincludethepuncturedPreparata-likecodesinthestate-
ment of Theorem 1 below, but this does not make any sense as there are no
Z Z -linear codes among them, see [7].
2 4
7 Results
Generally, a binary code can admit more than one Z Z structure. For ex-
2 4
ample, the 1-perfect code {0000000, 0001011, 0010110, 0101100, 1011000,
0110001, 1100010, 1000101, 1110100, 1101001, 1010011, 0100111, 1001110,
0011101, 0111010, 1111111} (this is the cyclic Hamming code of length 7,
see e.g. [18]) is closedwith respect to ∗ for 22 different involutions π, includ-
π
ing Id, (01)(24), (02)(14), and (04)(12) (and all their cyclic shifts). From the
characterisationofZ Z -linear1-perfectcodes,weseethata1-perfectcodeof
2 4
lengthatleast15oranextended1-perfectcodeoflengthmorethan16cannot
admittwo Z Z structureswithdifferent numberofself-adjacentcoordinates.
2 4
The next theorem, which is the main result of the paper, states more.
Theorem 1 Let C be a 1-perfect, extended 1-perfect, or Preparata-like code
of length n>16 closed with respect to both operations ∗ and ∗ , where π and
π τ
τ are Z Z structures. Then π =τ.
2 4
The theorem will be proven in the next section. Here, we consider some im-
portant corollaries.
Corollary 1 Let C be a 1-perfect, extended 1-perfect, or Preparata-like code
of length n>16. If C is Z Z -linear, i.e, closed with respect to the operation
2 4
∗ , for some Z Z structure π, then Sym(C)=Sym (C).
π 2 4 π
Proof Seeking a contradiction, assume that σ is from Sym(C) but not from
Sym (C).Thentheinvolutionτ d=ef σ−1πσ doesnotcoincidewithπ.However,
π
for any two codewords x and y,
x∗ y = x+y+(x+τ(x))·(y+τ(y))
τ
= σ−1 x′+y′+ x′+π(x′) · y′+π(y′)
(cid:0) (cid:0) (cid:1) (cid:0) (cid:1)(cid:1)
OntheAutomorphismGroupsofZ2Z4-LinearCodes 7
′ ′
belongstoC (herex =σ(x)∈C andy =σ(y)∈C).Wehaveacontradiction
with Theorem 1.
An exhaustive computer search shows that the statement of Corollary 1
holds also for the non-Z -linear extended 1-perfect codes of lengths 16 (as
4
follows,it is also true for the Z Z -linear 1-perfect codes of length 15, see the
2 4
observation in the second paragraph of the next section). For the Z -linear
4
extended1-perfectcodesoflength16,thesituationisdifferent.Oneofthetwo
non-equivalentcodes admits also the linear structure, andit is not difficult to
findthattheZ -linearstructureisnotpreservedbyallsymmetries.Theother
4
code meets |Sym(C)|=3|Sym (C)|. The order of the symmetry group of the
π
unique Preparata-likecode of length 16 is 16·15·14·12 [2], see also [4].
Corollary 2 (a) If C′ is a Z Z -linear 1-perfect code of type (2r −1,2t−1−
2 4
2r−1), t ≥ 4, t ≤ r ≤ t, then Sym(C′) is isomorphic to the automorphism
2
group of the group Zγ˙ ×Zδ, γ˙ d=ef 2r−t, δ d=ef t−r, and has the cardinality
2 4
γ˙ δ
221γ˙2−12γ˙+2γ˙δ+32δ2−12δY(2i−1)Y(2i−1).
i=1 i=1
(b) If C is a Z Z -linear extended 1-perfect code of type (2r,2t−1−2r−1),
2 4
t ≥ 4, t ≤ r ≤ t, then Sym(C) is isomorphic to a semidirect product of the
2
automorphism groupofthegroupZγ˙×Zδ,γ˙ d=ef 2r−t,δ d=ef t−r,withthegroup
2 4
Zγ˙+δ of translations by an element of order less that 4 and has the cardinality
2
γ˙ δ
221γ˙2+12γ˙+2γ˙δ+32δ2+21δY(2i−1)Y(2i−1).
i=1 i=1
(c) If C is a Z -linear extended 1-perfect code of rank 2t−r−1, t > 4,
4
t−1 ≤r ≤t−1, then Sym(C) has the cardinality
2
γ δ˙
212γ2+23γ+2γδ˙+23δ˙2+25δ˙+1Y(2i−1)Y(2i−1),
i=1 i=1
where γ d=ef 2r−t+1, δ˙ d=ef t−r−1.
Proof (b,c)Withoutlossofgenerality,wecanassumethattheZ Z structure
2 4
corresponding to the considered Z Z -linear code C has the form (1), where
2 4
α=2r in the case (b) and α=0 in the case (c).
ByCorollary1,Sym(C)=Sym (C)(thecaset=4iscoveredbythecom-
π
putational results mentioned above). Without loss of generality assume that
π is of form (1). Denote C d=ef Φ−1(C). According to (4), we have Sym (C) ≃
π
⊥
MAut(C). By Proposition 2, MAut(C) = MAut(C ). For a Z Z -linear ex-
2 4
⊥
tended 1-perfect code C, the related code C is a so-called Z Z -additive
2 4
Hadamardcode,see[19].Thestructureofthemonomialautomorphismgroup
8 DenisS.Krotov
of such codes was studied in [17], and statements (b) and (c) of the current
corollary follow from [17, Theorem 3] and [17, Theorem 2], respectively.
′
(a)Sincebyappending aparity-checkbit,everycodeC fromp.(a)results
′
inacodeC fromp.(b),wherethenewcoordinateisself-adjacent,Sym (C )co-
π
incideswiththestabilizerofaself-adjacentcoordinateinSym (C).Asfollows
π
⊥
directly from the structure of MAut(C ) considered in [17, Section IV] and
⊥
from the mentioned above connection between MAut(C ) and Sym (C), the
π
last groupcontains a subgroupisomorphic to Zr that acts transitively on the
2
2r self-adjacentcoordinates(inthe statement(b)ofthecurrentcorollary,this
subgroup is mentioned as the group of translations). By the orbit–stabilizer
theorem, we have |Sym (C′)|=|Sym (C)|/2r.
π π
Another interestingcorollaryfromTheorem1 wassuggestedby one ofthe
reviewers.RecallthattwoZ Z -linearorZ Z -additivecodesareequivalent if
2 4 2 4
oneofthecodescanbeobtainedfromtheotherbyamonomialtransformation,
that is, by a coordinate permutation and, if necessary, sign changes in some
coordinates.
Corollary 3 Let C and D be Z Z -additive codes such that C = Φ(C) and
2 4
D = Φ(D) are 1-perfect, extended 1-perfect, or Preparata-like codes. If C and
D are nonequivalent, then C and D are nonequivalent too.
Proof Both C and D are closed with respect to ∗ where π is in the form
π
(1). Assume that C and D are equivalent; i.e., C =σ(D) for some coordinate
permutation σ. Then, C is also closedwith respect to ∗σπσ−1. By Theorem 1,
we have π = σπσ−1. This means that σ preserves the pairs of adjacent coor-
dinates. It follows that Φ−1σΦ is a monomialtransformationand the codes D
and C =Φ−1(C)=Φ−1(σ(D))=Φ−1(σ(Φ(D))) are equivalent.
For the Z -linear Preparata-like codes, this fact is new, as their class has not
4
been completely characterized,in contrast to the case of (extended) 1-perfect
codes. It should be remarked, however, that the proof of Theorem 10.3(ii,iv)
in [9] stating the same as Corollary 3 for a partial class of Z -linear Prepara-
4
ta-like codes works for all Z -linear Preparata-like codes as well, taking into
4
account the later result [7] that only the Z -linear extended perfect code of
4
length 2t that has rank 2t−t can include a Z -linear Preparata-like code.
4
8 Proof of Theorem 1
Proof We first note that by Propositions 4 and 5, the statement on the Pre-
parata-like codes is straightforward from the one on the extended 1-perfect
codes. Indeed, the only extended 1-perfect code including a givenZ Z -linear
2 4
Preparata-like code P must be Z Z -linear with the same Z Z structure as
2 4 2 4
P.
Atsecond,appendingaparity-checkbittoeverycodewordofaZ Z -linear
2 4
′
1-perfectcodeC oftype(α,β)resultsinanextended1-perfectcodeC oftype
′
(α+1,β). Moreoveranysymmetry ofC is naturallyextendedto a symmetry
OntheAutomorphismGroupsofZ2Z4-LinearCodes 9
of C, which fixes the last (appended) coordinate. So, to prove the theorem, it
is sufficient to consider the case of an extended 1-perfect code C.
LetC be a Z Z -linearcode with twoZ Z structures,π andτ. Seeking a
2 4 2 4
contradiction,assumethatforsomecoordinatei,wehaveπ(i)6=τ(i).Without
′
loss of generality, π(i) 6= i. We will say that two coordinates j and j are
′ ′ ′ ′
independent if j 6∈{j,π(j),τ(j)} (equivalently, j 6∈{j ,π(j ),τ(j )}).
Suppose that C has two codewords v = (v0,...,vn−1) and u = (u0,...,
un−1) such that every nonzero coordinate of v is independent from every
nonzero coordinate of u, with the only exception v = u = 1. Then v∗ u
i π(i) τ
coincides with v+u (indeed, the situation in the middle bolded part of Ta-
ble1(a)neveroccursinthissum;infact,onlythefirstrowandthefirstcolumn
occur),whilev∗ udiffersfromv+uinthecoordinatesiandπ(i).Sinceboth
π
v ∗ u and v ∗ u must belong to C, we have a contradiction with the code
τ π
distance 4.
It remains to find such two codewords v, u. We restrict the search by the
weight-4codewords.Leti,i ,i ,i andπ(i),j ,j ,j betheonesofv andthe
2 3 4 2 3 4
onesofu,respectively.Itiseasytochoosei ,i ,i independentfromπ(i)and
2 3 4
tochoosej independentfromi,i ,i ,andi .Foreverychoiceofj ,thefourth
2 2 3 4 3
one j of u is defined uniquely. There are at least n−3·4−1 ways to choose
4
j independentfromi,i ,i ,andi .Inatleastn−3·4−1−3·4ofthem,the
3 2 3 4
resultingj isalsoindependentfromi,i ,i ,andi .Sincen−3·4−1−3·4≥
4 2 3 4
32−25>0, the result follows.
9 Acknowledgement
TheworkwassupportedbytheRussianFoundationforBasicResearch(grants
10-01-00424and13-01-00463).Theauthorwouldliketothankthe refereesfor
their valuable comments and suggestions.
References
1. Baker, R.D., van Lint, J.H., Wilson, R.M.: On the Preparata and Goethals codes.
IEEETrans.Inf.Theory29(3),342–345(1983). DOI10.1109/TIT.1983.1056675
2. Berlekamp, E.R.: Coding theory and the Mathieu groups. Inf.Control 18(1), 40–64
(1971). DOI10.1016/S0019-9958(71)90295-6
3. Bernal,J.J.,Borges,J.,Fern´andez-C´ordoba,C.,Villanueva,M.:Permutationdecoding
ofZ2Z4-linearcodes. Des.CodesCryptography 76(2), 269–277 (2015). DOI10.1007/
s10623-014-9946-4
4. Bierbrauer,J.:Nordstrom–RobinsoncodeandA7-geometry. FiniteFieldsAppl.13(1),
158–170(2007). DOI10.1016/j.ffa.2005.05.004
5. Borges,J.,Fern´andez-C´ordoba,C.,Pujol,J.,Rif`a,J.,Villanueva,M.:Z2Z4-linearcodes:
Generatormatricesandduality.Des.CodesCryptography54(2),167–179(2010). DOI
10.1007/s10623-009-9316-9
6. Borges,J.,Phelps,K.T.,Rif`a,J.:Therankand kernelof extended 1-perfectZ4-linear
and additive non-Z4-linear codes. IEEETrans.Inf.Theory 49(8), 2028–2034 (2003).
DOI10.1109/TIT.2003.814490
7. Borges, J., Phelps, K.T., Rif`a, J., Zinoviev, V.A.: On Z4-linear Preparata-like and
Kerdock-likecodes. IEEETrans.Inf.Theory49(11),3834–3843 (2003). DOI10.1109/
TIT.2003.819329
10 DenisS.Krotov
8. Borges, J., Rif`a, J.: A characterization of 1-perfect additive codes.
IEEETrans.Inf.Theory45(5),1688–1697 (1999). DOI10.1109/18.771247
9. Calderbank, A.R., Cameron, P.J., Kantor, W.M., Seidel, J.J.: Z4-Kerdock codes, or-
thogonal spreads, and extremal Euclidean line-sets 75(2), 436–480 (1997). DOI
10.1112/S0024611597000403
10. vanDam,E.R.,Fon-Der-Flaass,D.:Uniformlypackedcodesandmoredistanceregular
graphsfromcrookedfunctions. J.Algebr.Comb.12(2),115–121(2000). DOI10.1023/
A:1026583725202
11. Dumer,I.I.:Somenew uniformlypackedcodes. Proc.MoscowInst. PhysicsandTech-
nology.Ser.“Radiotekhnika iElektronika”,pp.72–78(1976). InRussian
12. Gorkunov, E.V.: The automorphism group of a q-ary Hamming code.
Diskretn.Anal.Issled.Oper.17(6),50–66(2010). InRussian
13. Hammons Jr,A.R.,Kumar,P.V.,Calderbank, A.R.,Sloane, N.J.A.,Sol´e, P.:TheZ4-
linearityofKerdock,Preparata,Goethals,andrelatedcodes. IEEETrans.Inf.Theory
40(2),301–319 (1994). DOI10.1109/18.312154
14. Kantor, W.M., Williams, M.E.: Symplectic semifield planes and Z4-linear codes.
Trans.Am.Math.Soc.356(3),895–938 (2004). DOI10.1090/S0002-9947-03-03401-9
15. Krotov, D.S.: Z4-linear perfect codes. Diskretn.Anal.Issled.Oper., Ser.1 7(4), 78–90
(2000). InRussian
16. Krotov, D.S.: Z4-linear perfect codes. E-print 0710.0198, ArXiv.org. URL
http://arxiv.org/abs/0710.0198.Englishtranslation
17. Krotov, D.S., Villanueva, M.: Classification of the Z2Z4-linear Hadamard codes and
their automorphism groups. IEEETrans.Inf.Theory 61(2), 887–894 (2015). DOI
10.1109/TIT.2014.2379644
18. MacWilliams,F.J.,Sloane,N.J.A.:TheTheoryofError-CorrectingCodes.Amsterdam,
Netherlands:NorthHolland(1977)
19. Phelps, K.T., Rif`a, J., Villanueva, M.: On the additive (Z4-linear and non-Z4-linear)
Hadamard codes: Rank and kernel. IEEETrans.Inf.Theory 52(1), 316–319 (2006).
DOI10.1109/TIT.2005.860464
20. Preparata, F.P.: A class of optimum nonlinear double-error correcting codes.
Inf.Control13(4),378–400(1968). DOI10.1016/S0019-9958(68)90874-7
21. Zaitsev,G.V.,Zinoviev,V.A.,Semakov,N.V.:InterrelationofPreparataandHamming
codes andextension of Hammingcodes to new double-error-correctingcodes. In: P.N.
Petrov,F.Csaki(eds.)Proc.2ndInt.Symp.InformationTheory,Tsahkadsor,Armenia,
USSR,1971,pp.257–264. AkademiaiKiado,Budapest, Hungary(1973)