

A085960


Size of the largest code of length 4 and minimum distance 3 over an alphabet of size n. This is usually denoted by A_{n}(4,3).


1



1, 2, 9, 16, 25, 34, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 484, 529, 576, 625, 676, 729, 784, 841, 900, 961, 1024, 1089, 1156, 1225, 1296, 1369, 1444, 1521, 1600, 1681, 1764, 1849, 1936, 2025, 2116, 2209, 2304, 2401, 2500
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OFFSET

1,2


COMMENTS

For n not 2 and not 6, a code C of size n^2 is given by two orthogonal Latin squares A and B of order n by C = {(i,j,A(i,j),B(i,j)): i,j in {1..n}}. Two orthogonal Latin squares of order n exist if and only if n is not 2 and not 6. See A055495.


REFERENCES

Raymond Hill, "A First Course in Coding Theory", Clarendon Press, Oxford, 1986 (see chapter 10, Theorem 10.16)


LINKS

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (3,3,1).


FORMULA

a(n) = 2 if n = 2, a(n) = 34 if n = 6, otherwise a(n) = n^2.
From Colin Barker, Nov 04 2015: (Start)
a(n) = 3*a(n1)3*a(n2)+a(n3) for n>9.
G.f.: x*(2*x^86*x^7+6*x^62*x^5+2*x^46*x^3+6*x^2x+1) / (x1)^3.
(End)


EXAMPLE

a(2) = 2 since the code C={0000,1110} has minimum distance 3 over the alphabet {0,1} and there is no such code with more codewords.


MATHEMATICA

Table[n^2  2 (Boole[n == 2] + Boole[n == 6]), {n, 50}] (* Wesley Ivan Hurt, Nov 04 2015 *)


PROG

(PARI) Vec(x*(2*x^86*x^7+6*x^62*x^5+2*x^46*x^3+6*x^2x+1)/(x1)^3 + O(x^100)) \\ Colin Barker, Nov 04 2015


CROSSREFS

Cf. A055495.
Sequence in context: A131189 A288484 A011193 * A051386 A003325 A101420
Adjacent sequences: A085957 A085958 A085959 * A085961 A085962 A085963


KEYWORD

easy,nonn


AUTHOR

W. Edwin Clark, Aug 17 2003


EXTENSIONS

More terms from David Wasserman, Feb 16 2005


STATUS

approved



