Best Known (65−37, 65, s)-Nets in Base 81
(65−37, 65, 370)-Net over F81 — Constructive and digital
Digital (28, 65, 370)-net over F81, using
- t-expansion [i] based on digital (16, 65, 370)-net over F81, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 16 and N(F) ≥ 370, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
(65−37, 65, 522)-Net over F81 — Digital
Digital (28, 65, 522)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8165, 522, F81, 37) (dual of [522, 457, 38]-code), using
- discarding factors / shortening the dual code based on linear OA(8165, 523, F81, 37) (dual of [523, 458, 38]-code), using
- 21 step Varšamov–Edel lengthening with (ri) = (2, 20 times 0) [i] based on linear OA(8163, 500, F81, 37) (dual of [500, 437, 38]-code), using
- extended algebraic-geometric code AGe(F,462P) [i] based on function field F/F81 with g(F) = 26 and N(F) ≥ 500, using
- 21 step Varšamov–Edel lengthening with (ri) = (2, 20 times 0) [i] based on linear OA(8163, 500, F81, 37) (dual of [500, 437, 38]-code), using
- discarding factors / shortening the dual code based on linear OA(8165, 523, F81, 37) (dual of [523, 458, 38]-code), using
(65−37, 65, 576445)-Net in Base 81 — Upper bound on s
There is no (28, 65, 576446)-net in base 81, because
- 1 times m-reduction [i] would yield (28, 64, 576446)-net in base 81, but
- the generalized Rao bound for nets shows that 81m ≥ 139 012036 131519 790913 318958 169874 044372 250874 309991 507043 060562 084552 317756 165358 814895 909039 607926 421420 730045 897419 349441 > 8164 [i]