7 encoding

Question 1

example 7.1
Let C be 3-ary and generated by
G =
[10010
01010
00102]
Encode the messagewords 000, 101 and 122.

Answer 1

000 → 00000, so we need
101 → (101)G = (10112)
122 → (122)G = (12201)

first three digits from this G give messageword digits
because G is in standard form

also the other digits are check digits

Question 2

why do we want G in standard form?

Answer 2

messageword encode
→ codeword transmit(noise) → received vector
project → nearest codeword interpret → decoded messageword

then drop the check digits

Question 3

For a given linear code C over F_q

(ie subset of F_q ^n for some n) generated by G we have a natural identification between C and F_q ^k
k=dimC

Answer 3

each x ∈ C is uniquely expressible as
x =Σ aᵢ vᵢ for i=1 to k

(the vᵢs are the rows of G in the natural order).

So
x ↔ (a1, a2, …, ak ) ∈ F_q ^k
is a one-to-one correspondence

Question 4

using G for messg words

Answer 4

We think of the a = (a1, …, ak ) vectors as the message words of
the code, and the n-tuples x as the codewords representing them.

Note that the encoding map
a→ x : F_q ^k → C
is then simply

x = aG
That is, right multiplication by the generating matrix — a linear map!