12 KiB
| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Bar product | 1/1 | https://en.wikipedia.org/wiki/Bar_product | reference | science, encyclopedia | 2026-05-05T14:39:58.994259+00:00 | kb-cron |
In information theory, the bar product of two linear codes C2 ⊆ C1 is defined as
C
1
∣
C
2
=
{
(
c
1
∣
c
1
+
c
2
)
:
c
1
∈
C
1
,
c
2
∈
C
2
}
,
{\displaystyle C_{1}\mid C_{2}=\{(c_{1}\mid c_{1}+c_{2}):c_{1}\in C_{1},c_{2}\in C_{2}\},}
where (a | b) denotes the concatenation of a and b. If the code words in C1 are of length n, then the code words in C1 | C2 are of length 2n. The bar product is an especially convenient way of expressing the Reed–Muller RM(d, r) code in terms of the Reed–Muller codes RM(d − 1, r) and RM(d − 1, r − 1). The bar product is also referred to as the | u | u + v | construction or (u | u + v) construction.
== Properties ==
=== Rank === The rank of the bar product is the sum of the two ranks:
rank
(
C
1
∣
C
2
)
=
rank
(
C
1
)
+
rank
(
C
2
)
{\displaystyle \operatorname {rank} (C_{1}\mid C_{2})=\operatorname {rank} (C_{1})+\operatorname {rank} (C_{2})\,}
==== Proof ==== Let
{
x
1
,
…
,
x
k
}
{\displaystyle \{x_{1},\ldots ,x_{k}\}}
be a basis for
C
1
{\displaystyle C_{1}}
and let
{
y
1
,
…
,
y
l
}
{\displaystyle \{y_{1},\ldots ,y_{l}\}}
be a basis for
C
2
{\displaystyle C_{2}}
. Then the set
{
(
x
i
∣
x
i
)
∣
1
≤
i
≤
k
}
∪
{
(
0
∣
y
j
)
∣
1
≤
j
≤
l
}
{\displaystyle \{(x_{i}\mid x_{i})\mid 1\leq i\leq k\}\cup \{(0\mid y_{j})\mid 1\leq j\leq l\}}
is a basis for the bar product
C
1
∣
C
2
{\displaystyle C_{1}\mid C_{2}}
.
=== Hamming weight === The Hamming weight w of the bar product is the lesser of (a) twice the weight of C1, and (b) the weight of C2:
w
(
C
1
∣
C
2
)
=
min
{
2
w
(
C
1
)
,
w
(
C
2
)
}
.
{\displaystyle w(C_{1}\mid C_{2})=\min\{2w(C_{1}),w(C_{2})\}.\,}
==== Proof ==== For all
c
1
∈
C
1
{\displaystyle c_{1}\in C_{1}}
,
(
c
1
∣
c
1
+
0
)
∈
C
1
∣
C
2
{\displaystyle (c_{1}\mid c_{1}+0)\in C_{1}\mid C_{2}}
which has weight
2
w
(
c
1
)
{\displaystyle 2w(c_{1})}
. Equally
(
0
∣
c
2
)
∈
C
1
∣
C
2
{\displaystyle (0\mid c_{2})\in C_{1}\mid C_{2}}
for all
c
2
∈
C
2
{\displaystyle c_{2}\in C_{2}}
and has weight
w
(
c
2
)
{\displaystyle w(c_{2})}
. So minimising over
c
1
∈
C
1
,
c
2
∈
C
2
{\displaystyle c_{1}\in C_{1},c_{2}\in C_{2}}
we have
w
(
C
1
∣
C
2
)
≤
min
{
2
w
(
C
1
)
,
w
(
C
2
)
}
{\displaystyle w(C_{1}\mid C_{2})\leq \min\{2w(C_{1}),w(C_{2})\}}
Now let
c
1
∈
C
1
{\displaystyle c_{1}\in C_{1}}
and
c
2
∈
C
2
{\displaystyle c_{2}\in C_{2}}
, not both zero. If
c
2
≠
0
{\displaystyle c_{2}\not =0}
then:
w
(
c
1
∣
c
1
+
c
2
)
=
w
(
c
1
)
+
w
(
c
1
+
c
2
)
≥
w
(
c
1
+
c
1
+
c
2
)
=
w
(
c
2
)
≥
w
(
C
2
)
{\displaystyle {\begin{aligned}w(c_{1}\mid c_{1}+c_{2})&=w(c_{1})+w(c_{1}+c_{2})\\&\geq w(c_{1}+c_{1}+c_{2})\\&=w(c_{2})\\&\geq w(C_{2})\end{aligned}}}
If
c
2
=
0
{\displaystyle c_{2}=0}
then
w
(
c
1
∣
c
1
+
c
2
)
=
2
w
(
c
1
)
≥
2
w
(
C
1
)
{\displaystyle {\begin{aligned}w(c_{1}\mid c_{1}+c_{2})&=2w(c_{1})\\&\geq 2w(C_{1})\end{aligned}}}
so
w
(
C
1
∣
C
2
)
≥
min
{
2
w
(
C
1
)
,
w
(
C
2
)
}
{\displaystyle w(C_{1}\mid C_{2})\geq \min\{2w(C_{1}),w(C_{2})\}}
== See also == Reed–Muller code
== References ==