IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 43, NO. 3, MAY 1997
969
Type II Codes over
Alexis Bonnecaze, Patrick Solé, Member, IEEE, Christine Bachoc, and Bernard Mourrain
Abstract— Type II 4 -codes are introduced as self-dual codes
over the integers modulo 4 containing the all-one vector and with
Euclidean weights multiple of 8. Their weight enumerators are
characterized by means of invariant theory. A notion of extremality for the Euclidean weight is introduced. Their binary images
under the Gray map are formally self-dual with even weights.
Extended quadratic residue 4 -codes are the main example of
this family of codes. They are obtained by Hensel lifting of the
classical binary quadratic residue codes. Their binary images
have good parameters. With every type II 4 -code is associated
via construction A modulo 4 an even unimodular lattice (type II
lattice). In dimension 32, we construct two unimodular lattices of
norm 4 with an automorphism of order 31. One of them is the
Barnes–Wall lattice BW32.
Index Terms—Codes over rings, lattices, self-dual codes, weight
enumerators, 4 -codes.
I. INTRODUCTION
T
HE conditions satisfied by the weight enumerator of selfdual codes, defined over the ring of integers modulo
four, have been studied by Klemm [1], then by Conway and
Sloane [2]. The MacWilliams transform determines a group
of substitutions, each of which fixes the weight enumerator of
a self-dual code. This weight enumerator belongs to the ring
of polynomials fixed by the group of substitutions, called the
ring
of invariants. Finding an explicit basis for this ring
is possible since is Cohen–Macaulay [3]. Each invariant is
written uniquely into this basis. (For more information about
this theory, see [4] and [5].)
Among all of the self-dual -codes, some have the property
that all Euclidean weights are multiples of . These codes
are called type II codes by analogy with the binary case. An
upper bound on their minimum Euclidean weight is given,
thereby leading to a natural notion of extremality akin to
similar concepts for type II binary codes and type II lattices.
The most interesting examples of type II codes are perhaps
the extended quadratic residue
-codes
. This class
of codes includes the octacode
and the lifted Golay
. Other classes of interest comprise a multilevel
construction from binary Reed–Muller and lifted double circulant codes. The paper is organized as follows. Section II
contains background information on -codes, the restrictions
on weight enumerators provided by invariant theory, and
recalls the results of Conway and Sloane in [2]. Section III
defines type II codes and gives the conditions satisfied by their
Manuscript received August 8, 1995; revised August 28, 1996.
A. Bonnecaze and P. Solé are with CNRS I3S, BP 145, 06903 Sophia
Antipolis, Cedex, France.
C. Bachoc is with the Laboratoire A2X, F-33405 Talence, France.
B. Mourrain is with INRIA, BP 93, 06902 Sophia Antipolis, Cedex, France.
Publisher Item Identifier S 0018-9448(97)02321-3.
complete weight enumerator. Section IV gives three examples
of families of type II codes with their general construction.
Section V shows the relation between type II codes and even
unimodular lattices.
II.
-CODES
AND INVARIANT
THEORY
By a -code of length
we shall mean a linear block
code over
that is an additive subgroup of
We define an
inner product on
by
,
and then the notions of dual code
, self-orthogonal code
, and self-dual code
are defined in the
standard way. We shall say that two -codes are equivalent
if one can be obtained from the other by permuting the
coordinates and (if necessary) changing the signs of certain
coordinates. The automorphism group
consists of all
monomial transformations (coordinate permutations and sign
changes) that preserve the set of codewords. We shall say that
a code is isodual if it is equivalent to its dual
Several weight enumerators are associated with a -code
The complete weight enumerator (or c.w.e.) of is
where
is the number of components of
that are
congruent to modulo . Since a monomial transformation
may change the sign of a component, the appropriate weight
enumerator for an equivalence class of codes is the symmetrized weight enumerator (or s.w.e.) given by
The MacWilliams identity over
expresses the weight enuin terms of the weight enumerator
merator of the dual code
of
which implies
The Gray map
provides a one-to-one correspondence
between a
-code and a binary code. Hammons et al. [8]
explored the Gray map which is a distance-preserving map
or isometry from
, Lee distance) to
, Hamming
distance). Recall that the Lee weights of the elements
0018–9448/97$10.00  1997 IEEE
970
of
are, respectively,
, and that the Lee weight of a
vector
is just the rational sum of the Lee weights of
its components. This weight function defines the Lee metric
on
We define maps
from
to
by
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 43, NO. 3, MAY 1997
where
and
is the ring of polynomials in
is the polynomial
This ring has the Molien series
and extend them in the obvious way to maps from
to
The Gray map
is given by
The binary image
of a -linear code
under the
Gray map need not be -linear, so that the dual code may
not even be defined. We define the -dual of
to be the
code
Thus
Example 3: The so-called lifted Golay
is a remarkable self-dual code over
It is an extended quadratic residue
-code. The expression of the symmetric weight enumerator
of the “lifted Golay” in terms of the basic polynomials given
in Theorem 2 is
note that the formal duality of the binary codes implies that
the diagram is noncommuting.
The binary image under the Gray map of a self-dual code
over
is formally self-dual (see [8]) and distance-invariant.
Furthermore, its weights are even, since the residue code
of its preimage is self-orthogonal. Using this simple
observation it follows immediately, using a well-known result
on binary type I codes (see [9, ch. 7]).
Theorem 1: The minimum Lee distance
of a self-dual
code of length
is at most
This bound is not tight in general (except for
in the
case of the octacode). It is a difficult open problem to sharpen
this bound using the following Gleason-type theorems.
In [1], Klemm has studied the conditions satisfied by the
complete weight enumerators of self-dual codes over
Conway and Sloane deduced analogous theorems for the
symmetrized and Hamming-weight enumerators [2, Theorems 6–9]. These results come from a “new application of
a nineteenth-century technique” [10]: invariant theory. The
weight enumerators are invariants of a group of substitutions.
The number of linearly independent homogeneous invariants
of degree is given by the coefficient of
in the Molien
series. The following theorem characterizes the symmetrized
weight enumerator of a self-dual code of length
over
containing a vector
Theorem 2: The symmetrized weight enumerator of a selfdual code of length over
containing a vector
belongs
to the ring
III. TYPE II
-CODES
A
-code of type II is a self-dual
-code containing a
vector
and which has the property that all Euclidean
weights are multiples of .
Theorem 4: The complete weight enumerator of a type II
code of length
over
belongs to the ring
where
is the ring of polynomials in
is the
of the octacode
is the
of
is the
of
is the
of the lifted Golay
where
is the
of
, the selfdual code introduced by Klemm [1], and
is a homogeneous
polynomial of degre
which does not belong to the algebra
This ring has Molien series
The complete weight enumerator of the code is left invariant
by the matrix group , of size
, generated by
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IV. EXAMPLES
with
OF
TYPE II CODES
We give three examples of families of type II codes with
their general construction.
A. Extended Quadratic Residue
where
is an th root of unity.
Remark 1: All the cwe of the codes we have considered
in this paper belong to the algebra
In degre ,
the dimension of this algebra is
and the dimension of the
invariant algebra is . The algebra generated by the
of
the type II codes is included into the invariant algebra, but we
do not know if we have equality.
Corollary 5: A -code of type II can only exist in lengths
multiple of .
Theorem 6: The symmetrized weight enumerator of a type
II code of length
over
belongs to the ring
where
is the ring of polynomials in
is the
of the octacode
is the
of
is the
of the lifted Golay
-Codes
Recently, Bonnecaze, Solé, and Calderbank investigated
extended quadratic residue
-codes [6]. They obtained the
parameters of the lifted Golay. These codes represent a remarkable class of type II. The image by the Gray map of
the octacode is the Nordstrom–Robinson and the image of the
lifted Golay is a
nonlinear binary code. In this
and
the extended quadratic
section, we describe
residue
-codes of length, respectively,
and . Recall
(see [6]) that a quadratic residue -code is defined to be the
cyclic code generated by
, the Hensel lift of the
generator polynomial
of the binary quadratic
residue code.
The code
is a
code with minimum Euclidean distance . It is generated by the polynomial
and extended by a parity check symbol. The complete weight
enumerator of
in terms of the basic polynomials is
where
is the
of
and
This ring has Molien series
The symmetric weight enumerator of the code is left invariant
by the matrix group , of size
, generated by
with
And the symmetrized weight enumerator of
the basic polynomials is
Its image by the gray map
is a
nonlinear formally self-dual code with a weight enumerator
symmetric with respect to
Number of words
where
is an th root of unity.
Remark 2: Results similar to Theorems 4 and 6 have been
obtained in [11, Theorem 4 and Corollary 5], with different
bases and, of course, the same Molien series.
in terms of
Weights
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IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 43, NO. 3, MAY 1997
In length , the binary extended quadratic residue code has
-code with the
very good parameters. It is a
following weight enumerator:
Remark 3: In [12], Pless and Qian give the explicit complete weight enumerator (without using invariant theory) of
and
B.
The Hensel lift of this code is a
by
-code generated
Its minimum Euclidean distance is and its symmetric weight
enumerator in terms of the basic polynomials is
Codes
denote
Let
Proposition 7: For
and
the code
is a type II -code.
Proof: The linearity of the code is trivial. Self-duality
follows by orthogonality of
and
To prove that all codewords have Euclidean weights divisible
by , write a generic codeword of
as
, with
and
Identifying binary
words with their supports, we get
By MacEliece’s theorem [4, p 447], for
is a
multiple of
(the case
is trivial).
By orthogonality of and
is even. Since
is also even. Hence, the Euclidean weight of
, that is,
is a multiple of .
Remark 4: The
of
is computed easily from the
joint weight enumerator of
and
(see [4, p. 149]).
Example 8:
is a type II code with parameters
and minimum Euclidean distance . The complete
weight enumerator of
in terms of the basic polynomials is
Its image by the Gray map
is a
nonlinear formally self-dual code with a weight enumerator symmetric with respect to
Number of words
Weights
Note that
is the Klemm code with parameters
.
The complete weight enumerator of
in terms of the basic
polynomials is
C. Double Circulant
-Codes
In [11], Calderbank and Sloane study the double circulant
codes
This code is similar to the code of [4 p. 507],
but it is read modulo four.
BONNECAZE et al.: TYPE II CODES OVER
to
Definition 9: Let
. The code
..
.
973
where is a prime congruent
has a generator matrix
..
.
if
Remark 5: This construction limits the minimum norm of
the lattice to four. To construct lattices with minimum norm
greater than four we should consider codes over
The metric generally used in lattice theory is the euclidean
metric. The theta series of
can be found by replacing
by
, or
..
.
..
.
if
, where
if
is a
nonzero square mod , or otherwise;
if
is not a square
, or otherwise; and can be
either
or
Theorem 10 (Calderbank and Sloane):
is a self-dual
code over
in which all norms are divisible by .
Example 11: For
, the code
is a type II code
with minimal norm .
V. EVEN UNIMODULAR LATTICES
An -dimensional lattice in
is the set of integer linear
combinations of
linearly independent vectors
The dual of an -dimensional lattice is denoted , and is
given by
, for all
A lattice
is integral if the inner product of any two lattice points is
integral, or equivalently, if
If is an integral lattice,
then
An integral lattice with
(or equivalently
is called unimodular. If
is an
even integer for all
then
is called even. The class
of even unimodular lattices includes the Gosset lattice
,
the Leech lattice
, and the Barnes–Wall
The theta
of the integral lattice gives the number of points
series
which are at equal distance from the origin. It is the formal
power series
where
is the number of vectors
with norm
The kissing number is the number of spheres touching one
sphere in the packing or equivalently the number of points on
the first layer of the lattice.
If
is a
-linear code with blocklength
then the
quaternary lattice
is given by
for some
in the symmetrized Lee weight enumerator of the -code
Theorem 12: Let
be a self-dual code over
such that
all Euclidean weights in
are divisible by . Then
is an even unimodular lattice.
.
Proof: See
Corollary 13: The minimum Euclidean distance
of a
type II -code of length
is at most
Proof: Let
be a type II
-code of length
The
lattice
being even unimodular by [6, Theorem 4.1],
its theta series
of weight
belongs to the polynomial
algebra
[9, ch. 7], where
is the theta series of the
root lattice
, and
Furthermore,
contains the norm
lattice
with theta series
If
, we have
With these notations, the smallest
such that
is nonzero
is
We then follow the same line as in [13]. The linear system
with unknowns
defined by the equations
has a unique solution
which is the theta series of the lattice associated to an extremal
code, if it exists.
The inequality
is equivalent to the
assumption
All the series are in
, and
Bürman’s formula shows that
where
which gives, using the fact that
series of the lattice
, where
is the theta
or equivalently
This is Construction A [9, ch. 5] applied to
it Construction A
-codes. We call
It is now sufficient to show that the coefficients of
are positive up to the index
(here
is the derivation of with respect to
).
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IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 43, NO. 3, MAY 1997
Let
be any type II
-code of length
is the lattice
Then
, such that
TABLE I
4 -CODES, THEIR BINARY IMAGES UNDER
THE
GRAY MAP
AND
ASSOCIATED LATTICES
and
Deriving
we find
We only have to show that
have positive coefficients up to
and
Proof: See [6].
Since
is a code of type II, the lattice obtained by the
same construction with the code
is an even unimodular
lattice. Furthermore, the minimum Euclidean weight of
is . Then, the produced lattice is of minimum norm and
its theta series is
and
Let
be a fixed integer. If
one can, by symmetry, put together the terms corresponding
to the different choices of One gets, as a coefficient of
which is positive as soon as
is greater than , because
is never zero.
A code meeting that bound with equality is called extremal.
The codes
and
are extremal and generate
extremal type II lattices via Construction A This is not
necessarily so. For instance,
is extremal even though the
lattice it generates contains vectors of norm by definition of
Construction A
Theorem 14: Let
and
be, respectively, the
Gosset, the Leech, and the Barnes–Wall lattice. Then
I.
II.
III.
IV.
Let us call this lattice
We see that
is
an even unimodular lattice of norm , therefore, extremal.
There are exactly two such lattices possessing furthermore an
automorphism of order . This is proved in [14]. One of them
is
It can be prooved [15] that
and
are nonequivalent. The lattice
can be constructed from
Construction A applied to the Reed–Muller -code of order
in the notation of [8]. This code has the same
as
(electronic computation due to Quian) and can
be considered as an extended lifted duadic code [16]. Both
constructions of
and
appear in [7, p. 226].
A. A Table of Codes and Lattices
The first two columns of Table I give the blocklength of the
code defined over , and the blocklength of the binary image
under the Gray map. The minimum distance (Lee distance in
world, Hamming distance in the
world) appears in
the
the third column. The
description of the code appears in
column 4, and the parameters of its binary image under the
Gray map appear in column 5 (here square brackets are used
to indicate a binary linear code). The last column describes
the lattice obtained from the -code by Construction A
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975
APPENDIX
COMPLETE WEIGHT ENUMERATORS
OF THE BASIC POLYNOMIALS
The
of
The
of
The
of
The
of
The
of the lifted Golay code
ACKNOWLEDGMENT
The authors wish to thank I. Duursma for helpful discussions
and A. R. Calderbank and N. J. A Sloane for sending them
a copy of [11], as well as V. Pless and Z. Qian for helpful
discussions and sending them a copy of [12]. More results on
type II -codes will appear in [17]. The authors also wish to
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IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 43, NO. 3, MAY 1997
thank the two anonymous referees for their detailed comments
on the manuscript.
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