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, 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

Offset: 1

Author

W. Edwin Clark, Aug 17 2003

Keywords

• easy
• nonn

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.

Examples

			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.
		

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).

Crossrefs

Cf. A055495.

Programs

Mathematica

Table[n^2 - 2 (Boole[n == 2] + Boole[n == 6]), {n, 50}] (* Wesley Ivan Hurt, Nov 04 2015 *)
LinearRecurrence[{3,-3,1},{1,2,9,16,25,34,49,64,81},50] (* Harvey P. Dale, Apr 18 2019 *)

PARI

Vec(-x*(2*x^8-6*x^7+6*x^6-2*x^5+2*x^4-6*x^3+6*x^2-x+1)/(x-1)^3 + O(x^100)) \\ Colin Barker, Nov 04 2015

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(n-1)-3*a(n-2)+a(n-3) for n>9.

G.f.: -x*(2*x^8-6*x^7+6*x^6-2*x^5+2*x^4-6*x^3+6*x^2-x+1) / (x-1)^3.

(End)

Extensions

More terms from David Wasserman, Feb 16 2005