Best Known (50−29, 50, s)-Nets in Base 81
(50−29, 50, 370)-Net over F81 — Constructive and digital
Digital (21, 50, 370)-net over F81, using
- t-expansion [i] based on digital (16, 50, 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
(50−29, 50, 386)-Net over F81 — Digital
Digital (21, 50, 386)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8150, 386, F81, 29) (dual of [386, 336, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(8150, 397, F81, 29) (dual of [397, 347, 30]-code), using
- 22 step Varšamov–Edel lengthening with (ri) = (4, 4 times 0, 1, 16 times 0) [i] based on linear OA(8145, 370, F81, 29) (dual of [370, 325, 30]-code), using
- extended algebraic-geometric code AGe(F,340P) [i] based on function field F/F81 with g(F) = 16 and N(F) ≥ 370, using
- 22 step Varšamov–Edel lengthening with (ri) = (4, 4 times 0, 1, 16 times 0) [i] based on linear OA(8145, 370, F81, 29) (dual of [370, 325, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(8150, 397, F81, 29) (dual of [397, 347, 30]-code), using
(50−29, 50, 361457)-Net in Base 81 — Upper bound on s
There is no (21, 50, 361458)-net in base 81, because
- 1 times m-reduction [i] would yield (21, 49, 361458)-net in base 81, but
- the generalized Rao bound for nets shows that 81m ≥ 3279 242415 226012 020633 985595 200106 421424 529646 706029 040683 412053 233757 346388 177509 570395 239361 > 8149 [i]