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A007229
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Number of distinct perforation patterns for deriving (v,b) = (n+2,n) punctured convolutional codes from (4,1).
(Formerly M5267)
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6
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38, 264, 2016, 15504, 122661, 986700, 8064576, 66756144, 558689224, 4719593312, 40193414112, 344721646640, 2974925353455, 25814778578820, 225105551191680, 1971557711151600, 17336058626562984, 152984380665537760
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OFFSET
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2,1
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COMMENTS
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"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). (End)
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REFERENCES
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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).
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LINKS
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FORMULA
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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) = (4,1), so
a(n) = (1/n)*Sum_{k|gcd(n+2,n)} phi(k)*binomial(4*n/k, (n+2)/k).
This simplifies to
a(n) = (1/n)*(binomial(4*n, n+2) + [(n mod 2) == 0]*binomial(2*n, (n/2) + 1)).
(Even though these formulas 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)
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MAPLE
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with(NumberTheory): P := proc(b, v0) local k: add(phi(k)*(1+z^k)^(v0*(b/k)), k=divisors(b))/b end: seq(coeff(P(b, 4), z, b+2), b=2..19); # Pab Ter
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MATHEMATICA
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P[b_, v0_] := Sum[EulerPhi[k]*(1+z^k)^(v0*(b/k)), {k, Divisors[b]}]/b;
Table[Coefficient[P[b, 4], z, b+2], {b, 2, 19}] (* Jean-François Alcover, Feb 17 2023, after Pab Ter's Maple code *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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More terms from Pab Ter (pabrlos2(AT)yahoo.com), Nov 13 2005
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STATUS
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approved
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