alx-ai/arxiv_papers · Datasets at Hugging Face

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H⋆=
\| \| \|
¯α1¯α2···¯αt
\| \| \|

Deﬁne, for ¯ µ∈C⋆,
K¯µ={(¯x1\|¯x2\|...\|¯xt\|¯x0)∈C: (σ1(¯x1),σ2(¯x2),...,σ(¯xt)) = ¯µ}.
Then,
C=/uniondisplay
¯µ∈C⋆K¯µ.
Further, since any two columns of H⋆are linearly independent, for any two distinct words
¯µand ¯µ′ofC⋆
d(K¯µ,K¯µ′)≥3. (3)
We will show that K¯µhas the properties given in Equation (1).
Any word ¯x= (¯x1\|¯x2\|...\|¯xt\|¯x0) must be at distance at most one from a word of
C, and hence, the word ( σ1(¯x1),σ2(¯x2),...,σ t(¯xt)) is at distance at most one from some
word ofC⋆. It follows that C⋆is a perfect code, and as a consequence, as C⋆is linear, it
is a Hamming code with parity-check matrix H⋆. As the number of rows of H⋆isr, we
then get that the number tof columns of H⋆is equal to
t=qr−1
q−1= 1+q+q2+...+qr−1.
3For any word ¯ x∗ofFn1+n2+...+ntq with ¯σ(¯x∗) = ¯µ∈C⋆, we now deﬁne the code C¯µ(¯x∗)
of lengthn0by
C¯µ(¯x∗) ={¯c∈Fn0
q: (¯x∗\|¯c)∈C}.
Again, using the fact that Cis a perfect code, we may deduce that for any ¯ x∗such
that the set C¯µ(¯x∗) is non empty, the set C¯µ(¯x∗) must be a perfect code of length n0=
(qs−1)/(q−1), for some integer s.
From the fact that the minimum distance of Cequals three, we get the property in
Equation (2).
Let ¯eidenote a word of weight one with the entry 1 in the coordinate positio ni. It
then follows that the two perfect codes C¯µ(¯x∗) andC¯µ(¯x∗+ ¯e1−¯ei), fori= 2,3,...,n 1,
must be mutually disjoint. Hence, n1is at most equal to the number of perfect codes in
a partition of Fn0qinto perfect codes, i.e.,
n1≤(q−1)n0+1 =qs.
Similarly,ni≤qs, fori= 2,3,...,t.
Reversing these arguments, using Equation (3) and the fact that Cis a perfect code,
we ﬁnd that ni, for eachi= 1,2,...,t, is at least equal to the number of words in an
1-ball ofFn0q.
We conclude that ni=qs, fori= 1,2,...,t, and ﬁnally
n=qs(1+q+q2+...+qr−1)+1+q+q2+...+qs−1= 1+q+q2+...+qr+s−1.
Givenr, we can then ﬁnd sfrom the equality
n= 1+q+q2+...+qm−1.
△
3. Combining construction of perfect codes
In the previous section, it was shown that a perfect code, depend ing on its rank, can
be divided onto small or large number of so-called ¯ µ-components, which satisfy some
equation with ¯ σ. The construction described in the following theorem realizes the ide a
of combining independent ¯ µ-components, diﬀerently constructed or taken from diﬀerent
perfect codes, in one perfect code.
A functionf: Σn→Σ, where Σ is some set, is called an n-ary(ormultary)quasigroup
of order \|Σ\|if in the equality z0=f(z1,...,z n) knowledge of any nelements of z0,z1,
...,znuniquely speciﬁes the remaining one.
Theorem 2. Letmandrbe integers, m>r,qbe a prime power, n= (qm−1)/(q−1)
andt= (qr−1)/(q−1). Assume that C∗is a perfect code in Ft
qand for every ¯µ∈C∗
we have a distance- 3codeK¯µ⊂Fn
qof cardinality qn−m−(t−r)that satisﬁes the following
generalized parity-check law:
¯σ(¯x) = (σ1(x1,...,x l),...,σ t(xlt−l+1,...,x lt)) = ¯µ
4for every ¯x= (x1,...,x n)∈K¯µ, wherel=qm−rand¯σ= (σ1,...,σ t)is a collections of
l-ary quasigroups of order q. Then the union
C=/uniondisplay
¯µ∈C∗K¯µ
is a perfect code in Fn
q.
Proof. It is easy to check that Chas the cardinality of a perfect code. The distance
at least 3 between diﬀerent words ¯ x, ¯yfromCfollows from the code distances of K¯µ(if
¯x, ¯ybelong to the same K¯µ) andC∗(if ¯x, ¯ybelong to diﬀerent K¯µ′,K¯µ′′, ¯µ′,¯µ′′∈C∗).△
The ¯µ-components K¯µcanbeconstructedindependentlyortakenfromdiﬀerentperfec t
codes. In the important case when all σiare linear quasigroups (e.g., σi(y1,...,y l) =
y1+...+yl) the components can be taken from any perfect code of rank at m ostn−r, as
followsfromtheprevioussection(itshouldbenotedthatif ¯ σislinear, thena ¯ µ-component
can be obtained from any ¯ µ′-component by adding a vector ¯ zsuch that ¯σ(¯z) = ¯µ−¯µ′).
In general, the existence of ¯ µ-components that satisfy the generalized parity-check law
for arbitrary ¯ σis questionable. But for some class of ¯ σsuch components exist, as we will
see from the following two subsections.
Remark. It is worth mentioning that ¯ µ-components can exist for arbitrary length tof
¯µ(for example, in the next two subsections there are no restriction s ont), if we do not
require the possibility to combine them into a perfect code. This is esp ecially important
for the study of perfect codes of small ranks (close to the rank o f a linear perfect code):
once we realize that the code is the union of ¯ µ-components of some special form, we may
forget about the code length and consider ¯ µ-components for arbitrary length of ¯ µ, which
allows to use recursive approaches.
3.1. Mollard-Phelps construction
Here we describe the way to construct ¯ µ-components derived from the product construc-
tion discovered independently in [7] and [9]. In terms of ¯ µ-components, the construction
in [9] is more general; it allows substitution of arbitrary multary quasig roups, and we will
use this possibility in Section 4.
Lemma 1. Let¯µ∈Ft
qand letC#be a perfect code in Fk
q. Letvandhbe(q−1)-ary
quasigroups of order qsuch that the code {(¯y\|v(¯y)\|h(¯y)) : ¯y∈Fq−1
q}is perfect. Let
V1, ...,VtandH1, ...,Hkbe respectively (k+1)-ary and (t+1)-ary quasigroups of order
q. Then the set
K¯µ=/braceleftBig
(¯x11\|...\|¯x1k\|y1\|¯x21\|...\|¯x2k\|y2\|...\|¯xt1\|...\|¯xtk\|yt\|z1\|z2\|...\|zk) :
¯xij∈Fq−1
q,

H⋆=

| | |

¯α1¯α2···¯αt

| | |



Deﬁne, for ¯ µ∈C⋆,

K¯µ={(¯x1|¯x2|...|¯xt|¯x0)∈C: (σ1(¯x1),σ2(¯x2),...,σ(¯xt)) = ¯µ}.

Then,

C=/uniondisplay

¯µ∈C⋆K¯µ.

Further, since any two columns of H⋆are linearly independent, for any two distinct words

¯µand ¯µ′ofC⋆

d(K¯µ,K¯µ′)≥3. (3)

We will show that K¯µhas the properties given in Equation (1).

Any word ¯x= (¯x1|¯x2|...|¯xt|¯x0) must be at distance at most one from a word of

C, and hence, the word ( σ1(¯x1),σ2(¯x2),...,σ t(¯xt)) is at distance at most one from some

word ofC⋆. It follows that C⋆is a perfect code, and as a consequence, as C⋆is linear, it

is a Hamming code with parity-check matrix H⋆. As the number of rows of H⋆isr, we

then get that the number tof columns of H⋆is equal to

t=qr−1

q−1= 1+q+q2+...+qr−1.

3For any word ¯ x∗ofFn1+n2+...+ntq with ¯σ(¯x∗) = ¯µ∈C⋆, we now deﬁne the code C¯µ(¯x∗)

of lengthn0by

C¯µ(¯x∗) ={¯c∈Fn0

q: (¯x∗|¯c)∈C}.

Again, using the fact that Cis a perfect code, we may deduce that for any ¯ x∗such

that the set C¯µ(¯x∗) is non empty, the set C¯µ(¯x∗) must be a perfect code of length n0=

(qs−1)/(q−1), for some integer s.

From the fact that the minimum distance of Cequals three, we get the property in

Equation (2).

Let ¯eidenote a word of weight one with the entry 1 in the coordinate positio ni. It

then follows that the two perfect codes C¯µ(¯x∗) andC¯µ(¯x∗+ ¯e1−¯ei), fori= 2,3,...,n 1,

must be mutually disjoint. Hence, n1is at most equal to the number of perfect codes in

a partition of Fn0qinto perfect codes, i.e.,

n1≤(q−1)n0+1 =qs.

Similarly,ni≤qs, fori= 2,3,...,t.

Reversing these arguments, using Equation (3) and the fact that Cis a perfect code,

we ﬁnd that ni, for eachi= 1,2,...,t, is at least equal to the number of words in an

1-ball ofFn0q.

We conclude that ni=qs, fori= 1,2,...,t, and ﬁnally

n=qs(1+q+q2+...+qr−1)+1+q+q2+...+qs−1= 1+q+q2+...+qr+s−1.

Givenr, we can then ﬁnd sfrom the equality

n= 1+q+q2+...+qm−1.

△

3. Combining construction of perfect codes

In the previous section, it was shown that a perfect code, depend ing on its rank, can

be divided onto small or large number of so-called ¯ µ-components, which satisfy some

equation with ¯ σ. The construction described in the following theorem realizes the ide a

of combining independent ¯ µ-components, diﬀerently constructed or taken from diﬀerent

perfect codes, in one perfect code.

A functionf: Σn→Σ, where Σ is some set, is called an n-ary(ormultary)quasigroup

of order |Σ|if in the equality z0=f(z1,...,z n) knowledge of any nelements of z0,z1,

...,znuniquely speciﬁes the remaining one.

Theorem 2. Letmandrbe integers, m>r,qbe a prime power, n= (qm−1)/(q−1)

andt= (qr−1)/(q−1). Assume that C∗is a perfect code in Ft

qand for every ¯µ∈C∗

we have a distance- 3codeK¯µ⊂Fn

qof cardinality qn−m−(t−r)that satisﬁes the following

generalized parity-check law:

¯σ(¯x) = (σ1(x1,...,x l),...,σ t(xlt−l+1,...,x lt)) = ¯µ

4for every ¯x= (x1,...,x n)∈K¯µ, wherel=qm−rand¯σ= (σ1,...,σ t)is a collections of

l-ary quasigroups of order q. Then the union

C=/uniondisplay

¯µ∈C∗K¯µ

is a perfect code in Fn

q.

Proof. It is easy to check that Chas the cardinality of a perfect code. The distance

at least 3 between diﬀerent words ¯ x, ¯yfromCfollows from the code distances of K¯µ(if

¯x, ¯ybelong to the same K¯µ) andC∗(if ¯x, ¯ybelong to diﬀerent K¯µ′,K¯µ′′, ¯µ′,¯µ′′∈C∗).△

The ¯µ-components K¯µcanbeconstructedindependentlyortakenfromdiﬀerentperfec t

codes. In the important case when all σiare linear quasigroups (e.g., σi(y1,...,y l) =

y1+...+yl) the components can be taken from any perfect code of rank at m ostn−r, as

followsfromtheprevioussection(itshouldbenotedthatif ¯ σislinear, thena ¯ µ-component

can be obtained from any ¯ µ′-component by adding a vector ¯ zsuch that ¯σ(¯z) = ¯µ−¯µ′).

In general, the existence of ¯ µ-components that satisfy the generalized parity-check law

for arbitrary ¯ σis questionable. But for some class of ¯ σsuch components exist, as we will

see from the following two subsections.

Remark. It is worth mentioning that ¯ µ-components can exist for arbitrary length tof

¯µ(for example, in the next two subsections there are no restriction s ont), if we do not

require the possibility to combine them into a perfect code. This is esp ecially important

for the study of perfect codes of small ranks (close to the rank o f a linear perfect code):

once we realize that the code is the union of ¯ µ-components of some special form, we may

forget about the code length and consider ¯ µ-components for arbitrary length of ¯ µ, which

allows to use recursive approaches.

3.1. Mollard-Phelps construction

Here we describe the way to construct ¯ µ-components derived from the product construc-

tion discovered independently in [7] and [9]. In terms of ¯ µ-components, the construction

in [9] is more general; it allows substitution of arbitrary multary quasig roups, and we will

use this possibility in Section 4.

Lemma 1. Let¯µ∈Ft

qand letC#be a perfect code in Fk

q. Letvandhbe(q−1)-ary

quasigroups of order qsuch that the code {(¯y|v(¯y)|h(¯y)) : ¯y∈Fq−1

q}is perfect. Let

V1, ...,VtandH1, ...,Hkbe respectively (k+1)-ary and (t+1)-ary quasigroups of order

q. Then the set

K¯µ=/braceleftBig

(¯x11|...|¯x1k|y1|¯x21|...|¯x2k|y2|...|¯xt1|...|¯xtk|yt|z1|z2|...|zk) :

¯xij∈Fq−1

q,