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A007227 Number of distinct perforation patterns for deriving (v,b) = (n+2,n) punctured convolutional codes from (3,1).
(Formerly M4624)
2
9, 42, 236, 1287, 7314, 41990, 245256, 1448655, 8649823, 52106040, 316360752, 1933910820, 11893566078, 73537906926, 456864894288, 2850557192175, 17854854154215, 112230508880490, 707714010205020, 4475876883386895 (list; graph; refs; listen; history; text; internal format)
OFFSET

2,1

COMMENTS

From Petros Hadjicostas, Jul 27 2020: (Start)

"A punctured convolutional code is a high-rate code obtained by the periodic elimination (i.e., puncturing) of specific code symbols from the output of a low-rate encoder. The resulting high-rate code depends on both the low-rate code, called the original code, and the number and specific positions of the punctured symbols." (The quote is from Haccoun and Bégin (1989).)

A high-rate code (v,b) (written as R = b/v) can be constructed from a low-rate code (v0,1) (written as R = 1/v0) by deleting from every v0*b code symbols a number of v0*b - v symbols (so that the resulting rate is R = b/v).

Even though the formulas below do not appear in the two published papers in the IEEE Transactions on Communications, from the theory in those two papers, it makes sense to replace "k|b" with "k|v0*b" (and "k|gcd(v,b)" with "k|gcd(v,v0*b)"). Pab Ter, however, uses "k|b" in the Maple program below. (End)

REFERENCES

Guy Bégin, On the enumeration of perforation patterns for punctured convolutional codes, Séries Formelles et Combinatoire Algébrique, 4th colloquium, 15-19 Juin 1992, Montréal, Université du Québec à Montréal, pp. 1-10.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

Table of n, a(n) for n=2..21.

Guy Bégin and David Haccoun, High rate punctured convolutions codes: Structure properties and construction techniques, IEEE Transactions on Communications 37(12) (1989), 1381-1385.

David Haccoun and Guy Bégin, High rate punctured convolutional codes for Viterbi and sequential coding, IEEE Transactions on Communications, 37(11) (1989), 1113-1125; see Section II.

FORMULA

From Petros Hadjicostas, Jul 27 2020: (Start)

The number of perforation patterns to derive high-rate convolutional code (v,b) (written as R = b/v) from a given low-rate convolutional code (v0, 1) (written as R = 1/v0) is (1/b)*Sum_{k|gcd(v,b)} phi(k)*binomial(v0*b/k, v/k).

According to Pab Ter's Maple code, this is the coefficient of z^v in the polynomial (1/b)*Sum_{k|b} phi(k)*(1 + z^k)^(v0*b/k).

Here (v,b) = (n+2,n) and (v0,1) = (3,1), so

a(n) = (1/n)*Sum_{k|gcd(n+2,n)} phi(k)*binomial(3*n/k, (n+2)/k).

This simplifies to

a(n) = (1/n)*(binomial(3*n, n+2) + [(n mod 2) = 0]*binomial(3*n/2, (n/2) + 1)), where [ ] is the Iverson bracket. (End)

MAPLE

with(numtheory):P:=proc(b, v0) local k: RETURN(add(phi(k)*(1+z^k)^(v0*(b/k)), k=divisors(b))/b): end; seq(coeff(P(b, 3), z, b+2), b=2..40); (Pab Ter)

CROSSREFS

Cf. A007223, A007224, A007225, A007229.

Sequence in context: A289364 A074443 A221425 * A224836 A195975 A185918

Adjacent sequences:  A007224 A007225 A007226 * A007228 A007229 A007230

KEYWORD

nonn

AUTHOR

Simon Plouffe

EXTENSIONS

More terms from Pab Ter (pabrlos2(AT)yahoo.com), Nov 13 2005

STATUS

approved

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Last modified August 9 15:33 EDT 2022. Contains 356026 sequences. (Running on oeis4.)