In information theory, the bar product of two linear codes C2 ⊆ C1 is defined as
![{\displaystyle C_{1}\mid C_{2}=\{(c_{1}\mid c_{1}+c_{2}):c_{1}\in C_{1},c_{2}\in C_{2}\},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/55a092c4a5bef90072a5d13b1c3f8478331179e7)
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[1]
or (u | u + v) construction.[2]
Properties[edit]
The rank of the bar product is the sum of the two ranks:
![{\displaystyle \operatorname {rank} (C_{1}\mid C_{2})=\operatorname {rank} (C_{1})+\operatorname {rank} (C_{2})\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a142c77af463421aa140ce221375b931f7e420d8)
Let
be a basis for
and let
be a basis for
. Then the set
is a basis for the bar product
.
Hamming weight[edit]
The Hamming weight w of the bar product is the lesser of (a) twice the weight of C1, and (b) the weight of C2:
![{\displaystyle w(C_{1}\mid C_{2})=\min\{2w(C_{1}),w(C_{2})\}.\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1e110b9984a8dc8ab092a964cae9d834043a693d)
For all
,
![{\displaystyle (c_{1}\mid c_{1}+0)\in C_{1}\mid C_{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0029bd1f7616ca0a06a4197d2dcc509352a86968)
which has weight
. Equally
![{\displaystyle (0\mid c_{2})\in C_{1}\mid C_{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a002a747a1fa27f85e17db10f91ac769451c093b)
for all
and has weight
. So minimising over
we have
![{\displaystyle w(C_{1}\mid C_{2})\leq \min\{2w(C_{1}),w(C_{2})\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/33577007766cdf632a81000c2caaf5f5be81f58e)
Now let
and
, not both zero. If
then:
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3c9cb5757c9637f588de3a96b0e66ba1022a9f8f)
If
then
![{\displaystyle {\begin{aligned}w(c_{1}\mid c_{1}+c_{2})&=2w(c_{1})\\&\geq 2w(C_{1})\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/37db82363f9407313ed46c0e88c39d2d8aabade8)
so
![{\displaystyle w(C_{1}\mid C_{2})\geq \min\{2w(C_{1}),w(C_{2})\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b5b1cbc09bbc65cef297f278683fbbd63552218)
See also[edit]
References[edit]