---
title: "Bar product"
chunk: 1/1
source: "https://en.wikipedia.org/wiki/Bar_product"
category: "reference"
tags: "science, encyclopedia"
date_saved: "2026-05-05T14:39:58.994259+00:00"
instance: "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 ==