Universal codes and unimodular lattices

R

OBIN

C

HAPMAN

P

ATRICK

S

OLÉ

Universal codes and unimodular lattices

Journal de Théorie des Nombres de Bordeaux, tome

8, n

o

_{2 (1996),}

p. 369-376

<http://www.numdam.org/item?id=JTNB_1996__8_2_369_0>

© Université Bordeaux 1, 1996, tous droits réservés.

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Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques

Universal Codes

and

Unimodular

Lattices

par ROBIN CHAPMAN ET PATRICK

SOLÉ

RÉSUMÉ. Les codes résidus

_quadratiques

binaires de

_longueur

p + 1

_produisent

_par construction B et

_bourrage

des réseaux de type II comme le réseau de Leech. _Récemment, il a été

_prouvé

_que les codes résidus

_{quadratiques quaternaires produisent}

les mêmes réseaux par construction A modulo 4. Nous montrons de manière directe l’

_équivalence

des deux constructions pour p ~ 31. En dimension 32 nous obtenons un réseau extrémal de _type II

_qui

n’est _pas isomètre au réseau de Barnes-Wall

_BW32.

On considère

également l’équivalence

entre construction B modulo 4

_plus

bour-rage et construction A modulo 8. En dimension 48 elles conduisent

toutes deux à une nouvelle

description

du réseau extrémal de _type

II

_appelé

_P48q.

ABSTRACT.

_{Binary quadratic}

residue codes of

_length

_{p + 1} pro-duce via construction B and

_{density doubling}

_type II lattices like the Leech.

_Recently,

_quaternary

_quadratic

residue codes have

been shown to

_produce

the same lattices

_by

construction A

mod-ulo 4. We prove in a direct _way the

_equivalence

of these two

con-structions for _{p ~} 31. In dimension

32,

we obtain an extremal lattice of _type II not isometric to the Barnes-Wall lattice

_BW32.

The

_equivalence

between construction B modulo 4

_plus

_density

doubling

and construction A modulo 8 is also considered. In di-mension 48

_they

both led to a new

_description

of the extremal

type II lattice

_P48q.

1. Introduction

In

_[2],

_Bonnecaze,

Sol6 and Calderbank introduce for

_primes

_{p - :1:1}

(mod 8),

codes 6

and

_R,

the universal extended

_quadratic

residue

_codes,

of

_length

_{p +} 1 over the 2-adic

_integers

_Z2°O.

For

_{positive integers s they}

consider their reductions

_Q23

and

N23

modulo

_2;

Q2

and

_N2

are

_just

the standard

_binary

extended

_quadratic

residue

_codes,

while

64

and

-A74

are the

quaternary

quadratic

residue codes. Given a code C of

_length

n over

_Z4

Mots-clés : _Quadratic residue codes, Lattices, construction _A, construction _{B, density}

dou-bling.

define

_A(C)

as the set of vectors in Z" which reduce modulo 4 to elements of C.

_{(mod 8)}

the lattice is even and unimodular

_([2]

Corollary

4.1);

if _p = 7 it is the

E8 lattice,

while

if p = 23

it is the Leech

lattice.

Here we

_show,

_by

means of an

_{explicit isomorphism,}

that if _{p ~ -1}

(mod 8)

and

_p

31 then is isometric to a lattice

L(Q2)

con-structed from the

_binary

_quadratic

residue code in a manner

(construction

B

_{plus density}

_{doubling ) generalizing}

the

_original

construction of the Leech lattice. If _p = 23 this

yields

a short

_proof

of what is

_perhaps

the

_simplest

construction of the Leech lattice

_[2].

_{If p = 31}

_this,

combined with results of Koch and

_Venkov,

shows that

BSBM32

introduced in

_[1]

is not isometric

to the Barnes-Wall lattice

_{BW32 .}

In section

_§4

we consider a

_quaternary

analogue

of this

_situation,

_replacing

construction B

_by

construction B mod-ulo

_4,

and construction A mod 4

_by

construction A mod 8. We

_show,

inter

alia,

that

_P48q

can be obtained in the latter _way from a

_quadratic

residue

code of

_length

48 over

Z8.

2. The main result

Throughout

this section we assume that _p is a

_prime

satisfying

_p = -1

(mod 8).

We also fix an

_integer

r such that r -1

(mod 4),

and

r2

+ _p = 0

(mod 32).

In addition if

_p

31 we will assume

that r2

_{+ p}

= 32.

(If p =

_7,

23 or

_31,

then r =

_5,

-3 _or 1

respectively.)

We first outline a construction of lattices from

_binary

codes of

_length

p + 1. Consider a

_{self-orthogonal}

linear subcode C of

Z2P+’,

_containing

the all-ones word. Define

_L(C)

to be the sublattice of

ZP+r

_generated

_by

the

following

types

of vectors:

(1)

all vectors of

_shape

_{(8 OP),}

(2)

all vectors of

_shape

₍₄₂

(3)

all vectors of

_shape

₍₂₁

_OP+1-1)

whose

_support

coincides with the

support

of an element of

C,

(4)

any vector of

_shape

_{(r 1P).}

This can be recast as the union of two cosets

of the lattice

_2B(C)

_obtained,

up to

scaling,

by

construction B

applied

to

C

_namely

with

_denoting

the

_parity-check

code of

_length

_{p +} 1. It is clear that

L (C)

is a

_lattice,

of index in If the code C is

_doubly

_even,

self-dual and

_doubly

even then the

lattice ~L(C)

is even and unimodular.

If C is

₆₂

or

R2

then it has these

_properties.

We

give

four

examples

of this

construction.

By

the norm of an element in Euclidean _space we mean the square of its

length,

and the minimal norm of a lattice is the least norm of a non-zero

el-ement of the lattice. It is easy to see that the minimum norm of

~L(~2)

is

where

_mw(C)

is the minimum

_(Hamming)

_weight

of the code C.

THEOREM 1. The lattices

~A(Q4)

and are isometric

_for

_p

31. Proof. Assume

_p

31. We recall the definition

of 6

from

_§III

of

_(2).

Let 6 be the _square root of _-p in

_Z2-

with 8 == -1

_{(mod 4).}

Note then that 6 - -r

(mod 16).

The vectors ma

(a

E

_Fp

U

_{oo})

are defined as the rows

of the matrix

(The

rows and columns of this matrix are labelled in the order

_{oo, 0, l, ... ,}

p -1.)

The matrix W is called a Jacobsthal

_{matrix ,}

and is instrumental in

building

Hadamard matrices of

_Paley

_type

_[10,

_Chap.

_II].

We collect here the

_properties

that we need

where J stands for the all-one matrix. See

_[10,

Chap.

II,

Lemma

_7]

for

proofs

of

_(Jl)

and

_(J2).

To _prove

_{(J3), (J4)}

observe

_firstly

that

_by

_(Jl)

Secondly, writing x

for the Jacobi

_symbol

we have

and

_by

the character

_property

_{of X}

the last

_{equality coming}

from

_(J2).

The coordinate

_positions

in the code are labelled

_{oo, 0,1, ... ,}

_{p -}

_1,

re-garded

as elements of the

_projective

line over

_F.

The universal extended

quadratic

residue code is now defined as

where

_Q2°O

is the field of 2-adic numbers. A similar definition holds for

N

with

_{replaced by}

We can now describe

A(04)

as the set of vectors in

_congruent

modulo 4 to elements of

~.

Let na E be the rows of the matrix

so that for a E

_Fp

U

_{oo}

we have _ma

(mod

16).

Since

_by

(J2)

we have NNt = 321 the matrix is

orthogonal.

We claim that this

matrix maps

_78L(jV2

to

_~A(Q4)’

This is

_equivalent

to

_saying

that N

maps to

A(Q4)’

Note that these codes and lattices are

preserved

by

the

_automorphism

Q

coming

from the

permutation

(0 1 2 ...

_{p -}

1)

on

_Fp

U

_{f oo},}

and this

_automorphism

maps ma to mOl+l and na to

This

_automorphism

is the shift in the

_cyclic

construction of the

_QR

codes. We

_proceed

to show that the

_{images by}

N of the four

_types

of vectors in construction L above lie in

11 ( C~4 ) .

Since the na E

_A(~4),

the matrix N takes the coordinate

_vectors,

which lie in into

A(~4).

For convenience let

_{(a, b; c; d)}

denote the

generic ,Q-coordinate

d where a and

0

are _any

quadratic residue,

and

quadratic

non-residue

_{respectively.}

Now

which lies in and is

_congruent

to modulo 8. Hence

+

_{mo) E}

A(Q4).

_Applying

Q it follows that +

_rrta)

E

A(~4)

for all a E

_Fp,

and

_{so 2 (rna}

+ E

A(Q4)

for all E

_Fp

U

_{oo}.

Hence

ivN

E

A(~4)

for all v of the

shape

(42 Op-1).

We next

_compute

where

_Q’

is the set of

_quadratic

non-residues modulo _p. The last coordinates estimates come from

(J3), (J4). Again

this has

_integer

_coordinates,

and is

congruent

modulo 4 +

_¿j=o

_mj),

so this vector lies in

A(C~4).

It follows that

I

/ B

for each k E

_FP.

But

_Nz

is

_{generated by}

the vectors whose

_supports

are

the sets U

_(k

+

_Q’).

_([2,

_p.370,

III.

_A.]).

It follows that if v has the

shape

(2a

and whose

_support

is the same as that of an element of

then E

A(Q4).

_Finally

and so e

A(Q4).

Hence C

_A(Q4)’

and

_comparing

determinants we see that =

A(Q4).

Since

L(Q2)

and

L(M)

_are

isometric the Theorem follows. D

3.

_Application

to the cases

of P =

23, 31.

If

_{(al, ... , an)}

is an element of a code over

Z4,

then its Euclidean

_weight

is

_w(al)

+ ... +

_w(an)

where

The minimum Euclidean

_weight

_mew(C)

of a code C over

_Z4

is the least Euclidean

_weight

of its non-zero elements. If C is a linear code then the

minimum norm of

A(C)

is

min(l6,mew(C)).

For p = 23 and _p =

31,

the

minimum norm of

L(92)

is

32,

and so the minimum norm of

A ( ~4)

is 16.

Hence

_mew(Q4)

&#x3E; 16. In

_[2]

this is

_proved

in a more elaborate _way for

p = 23.

In

_[8]

Koch and Venkov show that for the five

_{non-isomorphic doubly}

even

self-dual

_binary

codes

_{Cl, ... , C5}

of

_length

_32,

the lattices

_{L(Cl), ... , L(C5)}

are all non-isometric. We can take

01

=

Q2,

and

C2

_to be the Reed-Muller

code

_{RM(2, 5).}

Since

_{L(RM(2, 5))}

is isometric to the Barnes-Wall lattice

BW32

_[9],

it follows that

_tA(Q4)

for _p = 31 is not isometric _to

BW3z,

con-firming

a

_conjecture

of

[1].

It is known that there are

only

two unimodular lattices in dimension 32 with minimal norm 4 and an

_automorphism

of order

31

_[12].

From the results of

_[1]

and of the current _paper we can infer than

both can be constructed

by

construction A mod 4

applied

to an extended

quaternary

cyclic

code: the

_quaternary

Reed-Muller code

_{QRM(2, 5)}

in

the case of

BW32

and the extended

quadratic

residue code

64

in the case

of

_BSBM32

:= Both lattices also _{appear in}

[11, 4].

4.

_{Quaternary Analogue}

We assume in

_{this §}

that

_{p &#x3E;}

47 is a

_{p e =-}

-1

(mod 8),

and that the

integer

r - 1

(mod

4)

satisfies

if _{p =}

_{47, 71}

and

if p =

79,103,127.

The

_{corresponding}

values of r are r =

-7,5

in first _case

and r

_{= -7, 5,1}

in the second. For a

_quaternary

code C of

_length

_p + 1 we define

-and

For an

_octonary

code

_C8

of

_length

_p

_{+ 1,}

we define

We have the

_following

_analogue

of Theorem 1:

THEOREM 2. The lattices

)L4 (1ij4)

and

_{-1-A4 (~8)}

are

_{isometric for p}

=

47,

71, 79, 103,

127.

The

_proof

is

_analogous

to the

_proof

of Theorem 1 and is omitted. COROLLARY 1. For p = 47 the lattice has _norm

_6,

and the code

Proof. Follows from the

_preceding

theorem

_{by noticing}

that

~4

has

eu-clidean minimum

_weight

24

_{[1, 11, 5].}

0

The lattice

L4 (~4)

was considered in

[3]

and is isometric to

P48q.

Adopt-ing

the definition of

_P48q

in

_§7.7

of

_[6],

the

_orthogonal

matrix

takes

_P48q

to

_{L4 (N~)}

_(which

is isometric to

L4 (~4))

_by

a similar

_argument

to Theorem 1.

_Similarly

it is tantamount to

_conjecture

that the

_conjectural

extremal

_type

II lattice of dimension 80 of

_example

3 of

_[13]

is taken

_by

into

_~4(~/4).

5. Conclusion

It would be

_interesting

to lift the

_remaining

three

_Conway-Pless

codes

over

_Z4

and obtain

_by

construction

A4

the three

_remaining

zero-defect

lat-tices of the Koch-Venkov classification.

_Similarly

the construction of

_P48q

by

construction

B3

applied

to

_ternary

_QR

codes and

_density

_doubling

_[6,

p.149]

suggests

a construction modulo 6.

Eventually,

_quaternary

double circulant codes which

_produce

an even extremal unimodular lattice in di-mension 40

_[5]

should be amenable to a similar

analysis.

6.

_{Acknowledgments}

The work described in this paper was done when both authors were

Visiting

Fellows at the School of

Mathematics,

Physics,

Computing

and Electronics at

_{Macquarie University.}

The first author would like to thank

Gerry Myerson

for his kind invitation. The second author would like to

thank Peter Pleasants for his kind invitation and

_Macquarie

_University

for

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Vol. 60

₍₁₉₉₃₎

40-65. Robin CHAPMAN

Department

of Mathematics

University

of Exeter

EX4 4QE

U.K.

rjc@maths.exeter.ac.uk

Patrick SOLI CNRS-I3S BP 145

06903

_Sophia

_Antipolis

cedex France