Definition of a convolutional MCS code.
Let, for some integer value N, N > 3, such that N + 1 is a prime number, there is some fixed sequence AM of length M, M > 1, of elements from Z/N,
AM = {a1,a2,…,aM}, (3.2.1)
ai ϵ Z/N for all i ϵ {1,2,…,M}.
The sequence (3.2.1) will henceforth be called the information sequence.
Everywhere below, the operations of addition and subtraction are implied in the ring Z/N.
Let B2M = {b1,b2,…,b2M} be a sequence of elements from Z/N of length 2M, composed of (3.2.1) by alternating signs of the information sequence and zeros:
B2M = {b1,b2,…,b2M} = {a1,0,a2,0,…,am,0}. (3.2.2)
In this way,
b2i - 1 = ai, b2i = 0
for all i ϵ {1,2,…,M}.
Sequence (3.2.2) will be referred to as a plaintext sequence in which information values are in odd places and zeros are in even places.
Let π be some logarithmic substitution from SN.
Let
E2M = {e1,e2,…,e2M} (3.2.3)
a sequence of elements from Z/N in which
e1 = b1, e2 = b2 + π(e1), ei = bi + π(ei-2 + ei-1) - π(ei-1) (3.2.4)
for all i ϵ {3,4,…,2M}.
The sequence (3.2.3) will henceforth be called the encoded plaintext sequence (3.2.2), or the communication channel sequence. The procedure (3.2.4) of obtaining (3.2.3) itself will be called the encoding process.
From (3.2.4) it follows that
b1 = e1, b2 = e2 - π(e1), bi = ei - (π(ei-2 + ei-1) - π(ei-1)) (3.2.5)
The calculation of the plaintext using (3.2.5) will be called the decoding of the communication channel sequence (3.2.3) into plaintext. When decoding the sequence of the communication channel in the received plaintext, the information values must alternate with zeros.
Thus, as follows from (3.2.2), (3.2.3) and (3.2.4), when encoding, the size of the encoded sequence increases exactly 2 times compared to the size of the original information sequence.
Unlike most known convolutional codes, the convolutional MCS code performs all operations not with bits, but with elements from the Z/N ring, where N+1 is a prime number. Important examples of such values of N are N = 16 and N = 256. In the latter case, all operations are performed on bytes.
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