Bar product - Alchetron, The Free Social Encyclopedia

In information theory, the bar product of two linear codes C₂ ⊆ C₁ is defined as

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 )

Let { x 1 , … , x k } be a basis for C 1 and let { y 1 , … , y l } be a basis for C 2 . Then the set

{ ( x i ∣ x i ) ∣ 1 ≤ i ≤ k } ∪ { ( 0 ∣ y j ) ∣ 1 ≤ j ≤ l }

is a basis for the bar product C 1 ∣ C 2 .

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 ) } .

For all c 1 ∈ C 1 ,

( c 1 ∣ c 1 + 0 ) ∈ C 1 ∣ C 2

which has weight 2 w ( c 1 ) . Equally

( 0 ∣ c 2 ) ∈ C 1 ∣ C 2

for all c 2 ∈ C 2 and has weight w ( c 2 ) . So minimising over c 1 ∈ C 1 , c 2 ∈ C 2 we have

w ( C 1 ∣ C 2 ) ≤ min { 2 w ( C 1 ) , w ( C 2 ) }

Now let c 1 ∈ C 1 and c 2 ∈ C 2 , not both zero. If c 2 ≠ 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 )

If c 2 = 0 then

w ( c 1 ∣ c 1 + c 2 ) = 2 w ( c 1 ) ≥ 2 w ( C 1 )

so

w ( C 1 ∣ C 2 ) ≥ min { 2 w ( C 1 ) , w ( C 2 ) }

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