Best Known (39, 78, s)-Nets in Base 81
(39, 78, 730)-Net over F81 — Constructive and digital
Digital (39, 78, 730)-net over F81, using
- t-expansion [i] based on digital (36, 78, 730)-net over F81, using
- net from sequence [i] based on digital (36, 729)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 36 and N(F) ≥ 730, using
- the Hermitian function field over F81 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 36 and N(F) ≥ 730, using
- net from sequence [i] based on digital (36, 729)-sequence over F81, using
(39, 78, 2120)-Net over F81 — Digital
Digital (39, 78, 2120)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8178, 2120, F81, 3, 39) (dual of [(2120, 3), 6282, 40]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(8178, 2189, F81, 3, 39) (dual of [(2189, 3), 6489, 40]-NRT-code), using
- OOA 3-folding [i] based on linear OA(8178, 6567, F81, 39) (dual of [6567, 6489, 40]-code), using
- construction X applied to C([0,19]) ⊂ C([0,18]) [i] based on
- linear OA(8177, 6562, F81, 39) (dual of [6562, 6485, 40]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 814−1, defining interval I = [0,19], and minimum distance d ≥ |{−19,−18,…,19}|+1 = 40 (BCH-bound) [i]
- linear OA(8173, 6562, F81, 37) (dual of [6562, 6489, 38]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 814−1, defining interval I = [0,18], and minimum distance d ≥ |{−18,−17,…,18}|+1 = 38 (BCH-bound) [i]
- linear OA(811, 5, F81, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(811, s, F81, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,19]) ⊂ C([0,18]) [i] based on
- OOA 3-folding [i] based on linear OA(8178, 6567, F81, 39) (dual of [6567, 6489, 40]-code), using
- discarding factors / shortening the dual code based on linear OOA(8178, 2189, F81, 3, 39) (dual of [(2189, 3), 6489, 40]-NRT-code), using
(39, 78, 5376645)-Net in Base 81 — Upper bound on s
There is no (39, 78, 5376646)-net in base 81, because
- 1 times m-reduction [i] would yield (39, 77, 5376646)-net in base 81, but
- the generalized Rao bound for nets shows that 81m ≥ 898 145569 592605 914803 285456 369128 549622 658320 501263 663840 617997 356616 219017 359535 495378 196197 054762 791076 587722 225355 798610 069659 246363 378054 556321 > 8177 [i]