7.3 Synthesis of a cyclic code combination
The cyclic code combination can be received in two ways. The first turns out multiplication of information sequence to generating polynomials Р(х), that leads to formation of an inseparable cyclic code. Inseparability considerably complicates decoding process, therefore in practice use the second way at which the information sequence is multiplied by a monomial хr is more often and the remainder of division of the received sequence on generating polynomials is added. It can be written down in the form of the formula:
(7)
Where F (x) – a cyclic code combination;
G (x) – information sequence in polynomial form;
- remainder
of division on generating polynomials.
For transfer of binary sequence in polynomial form each bit (1 or 0) is multiplied on х in the degree corresponding to a site of this bit.
Let's translate the sequence received in item 7.1 in polynomial form.
S |
I |
P |
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0 |
1 |
0 |
0 |
0 |
0 |
1 |
1 |
0 |
0 |
1 |
1 |
1 |
0 |
0 |
1 |
0 |
1 |
0 |
0 |
0 |
0 |
0 |
0 |
х23 |
х22 |
х21 |
х20 |
х19 |
х18 |
х17 |
х16 |
х15 |
х14 |
х13 |
х12 |
х11 |
х10 |
х9 |
х8 |
х7 |
х6 |
х5 |
х4 |
х3 |
х2 |
х1 |
х0 |
The received code combination can be written down as:
G (x) = х22 + х17 + х16 + х13 + х12 + х11 + х8 + х6.
Let's increase G (x) by a monomial хr. As the quantity of verifying categories calculated in item 7.2 equally seven, it is multiplied on х7
G (x) х7 = х29 + х24 + х23 + х20 + х19 + х18 + х15 + х13.
For reception of the resolved combination of a cyclic code we will divide the received sequence into generating polynomials chosen in item 4.2. Division process is shown more low.
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x7+x4+x3+1 |
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x22+x19+x18+x17++x15+x12+x11+x10+x9+x6+x4+x3+x |
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x6+x5+x4+x3+x+1 = R (x) |
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So, the resolved cyclic code combination, according to the formula (7) looks like:
F (x) = x29+x24+x23+x20+x19+x18+x15+x13+ x6+x5+x4+x3+x+1.
Let's translate it in a binary kind:
100001100111001010000001111011