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:= rxx + b .
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16: Cyclic Codes
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P,r := 0,0 ;
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{ Invariant: degree.r < degree.Q A (3d :: P = Qxd + r} } do true get.b { O^b^ I } ;
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P := Pxx + b ;
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r := rxx + b + Qxrm-i ', put.r
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Figure 16.7 On-line computation of cyclic codes. This increases the degree of r by one. Consequently, it may falsify degree.r < degree.Q . If it does, the degree of r after the assignment must be equal to the degree of Q. The invariant can thus be re-established by adding Q to r . If it does not, no further action needs to be taken. So the assignments to r that we have to implement are
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r := rxx + b ;
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if degree.r < degree.Q n degree.r ^ degree.Q skip
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r := r + Q
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A direct hardware implementation of the conditional statement would be inefficient. However, by observing that the assignment r := rxx + b falsifies degree.r < degree.Q exactly when rm-\ = 1, we see that it may be replaced by the unconditional assignment r := r + Qxr m _i . The two assignments can now be combined resulting in the program in Figure 16.7. The assignment to r does, indeed, have the form of the assignment in (16.2). (The 'modx m ' can be dropped because the resulting value of r is guaranteed to have degree less than m.) The coefficient, c, in (16.2) takes the value r m _i, the bit of the remainder that is 'shifted out' of the shift register by the operation r := rxx + b. So, the assignment tor can be implemented by a simple feedback loop. Figure 16.8 illustrates the circuit for the particular generator polynomial
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16.6 Summary
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Figure 16.8 Remainder computation with generator Q =
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Exercise 16.3. Earlier on, we said that what is transmitted is the remainder after dividing xmxP by Q, where P is the data polynomial. This means that the input polynomial is terminated by m zeros. A direct implementation in hardware of the program in Figure 16.7 would therefore involve a delay of m steps inputting the trailing zeros before the remainder could be output. Develop a program that eliminates this undesirable delay by taking as invariant the property degree.r < degree.Q A (3d :: xmxP = Q_xd
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Cyclic codes are used to add redundancy to transmitted data in order to detect and/or repair transmission errors. Computation of cyclic codes involves remainder computation in an algebra of polynomials. Two algorithms have been discussed for computing cyclic codes. One is similar to long division in integer arithmetic, and the second is an on-line algorithm suitable for direct implementation in hardware.
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Bibliographic Remarks
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More information on cyclic codes can be found in (for example) Blahut (1983).
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Appendix
This appendix contains a summary of the mathematical laws discussed in the main text.
Prepositional Calculus
Minimal Basis. A minimal basis for the prepositional calculus comprises three operators ('logical connectives'), listed below in ascending order of precedence, equivalence (=) , disjunction (v) , negation (-1) , and the following laws. Associativity of =: Symmetry of =: Unit of =: Negation: ((p = q) = r) = (p = (q = r)) . p=q=q=p .
true = p~p . ->p = p = false . ->(p = q) = - > p ^ q .
Distributivity of ->: Symmetry ofv: Associativity ofv: Idempotence of v: Distributivity ofv: Excluded middle:
pvq=qvp . ( p v q)vr = pv (qvr) . pvp = p . pv(q = r) = pvq = pvr . pv-^p .
Note that, in all but the first law, the associativity of equivalence is assumed.
Appendix
Additional Operators. The remaining logical connectives are conjunction (A) ,
if(<=) ,
only if (=>) , inequivalence (^) . These are defined in terms of equivalence, disjunction and negation in the following laws. The precedence convention is that conjunction has the same precedence as disjunction, and inequivalence has the same precedence as equivalence. 'If and 'only if have the same precedence, which is less than conjunction and disjunction, and more than equivalence and inequivalence. false: false = p /\q = p = q = pv q . (p q) = ~^(p = q) .
Golden rule: Inequivalence: If: