A007225 - OEIS

A007225

Number of distinct perforation patterns for deriving (v,b) = (n+4,n) punctured convolutional codes from (2,1).
(Formerly M2023)

2, 12, 52, 232, 952, 3888, 15504, 61333, 240350, 937508, 3641820, 14112560, 54587280, 210907168, 814278240, 3142611402, 12126758436, 46796872472, 180619420520, 697320058864, 2693097842512, 10405151052320, 40219629005920

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OFFSET

5,1

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=5..27.

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+4,n) and (v0,1) = (2,1), so

a(n) = (1/n)*Sum_{k|gcd(n+4,n)} phi(k)*binomial(2*n/k, (n+4)/k).

This simplifies to

a(n) = (1/n)*(binomial(2*n, n+4) + [(n mod 2) == 0]*binomial(n, (n/2) + 2) + 2*[(n mod 4) == 0]*binomial(n/2, (n/4) + 1)). (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, 2), z, b+4), b=5..40); (Pab Ter)

MATHEMATICA

a[n_] := DivisorSum[GCD[n+4, n], EulerPhi[#]*Binomial[2n/#, (n+4)/#]&]/n;

Table[a[n], {n, 5, 27}] (* Jean-François Alcover, Mar 10 2023 *)

CROSSREFS

Cf. A007223, A007224, A007229.