Best Known (72−31, 72, s)-Nets in Base 16
(72−31, 72, 524)-Net over F16 — Constructive and digital
Digital (41, 72, 524)-net over F16, using
- trace code for nets [i] based on digital (5, 36, 262)-net over F256, using
- net from sequence [i] based on digital (5, 261)-sequence over F256, using
(72−31, 72, 675)-Net over F16 — Digital
Digital (41, 72, 675)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1672, 675, F16, 31) (dual of [675, 603, 32]-code), using
- 27 step Varšamov–Edel lengthening with (ri) = (3, 1, 0, 1, 6 times 0, 1, 16 times 0) [i] based on linear OA(1666, 642, F16, 31) (dual of [642, 576, 32]-code), using
- trace code [i] based on linear OA(25633, 321, F256, 31) (dual of [321, 288, 32]-code), using
- extended algebraic-geometric code AGe(F,289P) [i] based on function field F/F256 with g(F) = 2 and N(F) ≥ 321, using
- trace code [i] based on linear OA(25633, 321, F256, 31) (dual of [321, 288, 32]-code), using
- 27 step Varšamov–Edel lengthening with (ri) = (3, 1, 0, 1, 6 times 0, 1, 16 times 0) [i] based on linear OA(1666, 642, F16, 31) (dual of [642, 576, 32]-code), using
(72−31, 72, 214368)-Net in Base 16 — Upper bound on s
There is no (41, 72, 214369)-net in base 16, because
- 1 times m-reduction [i] would yield (41, 71, 214369)-net in base 16, but
- the generalized Rao bound for nets shows that 16m ≥ 31 084372 790641 041903 092167 610684 495591 002712 036122 536481 303995 759831 930125 864012 471776 > 1671 [i]