Coding Theory: Let C₁ be a binary (n, M₁, d₁)-code and C₂ be a binary (n, M₂, d2)-code. Consider the binary code C3 := {(u | u + v) | u € C₁, v E C₂} . Note: (u Ju+v) denotes the concatenation of the word u and the word u+v. Example: if u = (U₁, U₂..., Un) and u+v = (W₁, . . . , Wn) then (u |u+v) = (u₁ . . ., Un, W₁, . . . , Wn)) ..., Prove that: - the length of C3 is 2n the number of codewords is M₁M2 the distance in otly d
Coding Theory: Let C₁ be a binary (n, M₁, d₁)-code and C₂ be a binary (n, M₂, d2)-code. Consider the binary code C3 := {(u | u + v) | u € C₁, v E C₂} . Note: (u Ju+v) denotes the concatenation of the word u and the word u+v. Example: if u = (U₁, U₂..., Un) and u+v = (W₁, . . . , Wn) then (u |u+v) = (u₁ . . ., Un, W₁, . . . , Wn)) ..., Prove that: - the length of C3 is 2n the number of codewords is M₁M2 the distance in otly d
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:Coding Theory:
Let C₁ be a binary (n, M₁, d₁)-code and C₂ be a binary (n, M2, d2)-code. Consider the
binary code C3 := {(u | u + v) | u € C₁, v E C₂} .
Note: (u Ju+v) denotes the concatenation of the word u and the word u+v.
Example: if u = (U₁, U₂..., Un) and u+v = (W₁, . . . , Wn)
then (u |u+v) = (u₁ . . ., Un, W₁, . . . , Wn))
Prove that:
-
-
the length of C3 is 2n
the number of codewords is M₁M₂
the distance is exactly d3
Hence:
C3 is a binary (2n, M₁M2, d3)-code
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