Best Known (40, 40+38, s)-Nets in Base 81
(40, 40+38, 730)-Net over F81 — Constructive and digital
Digital (40, 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
(40, 40+38, 2406)-Net over F81 — Digital
Digital (40, 78, 2406)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8178, 2406, F81, 2, 38) (dual of [(2406, 2), 4734, 39]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(8178, 3286, F81, 2, 38) (dual of [(3286, 2), 6494, 39]-NRT-code), using
- OOA 2-folding [i] based on linear OA(8178, 6572, F81, 38) (dual of [6572, 6494, 39]-code), using
- construction X applied to Ce(37) ⊂ Ce(33) [i] based on
- linear OA(8175, 6561, F81, 38) (dual of [6561, 6486, 39]-code), using an extension Ce(37) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,37], and designed minimum distance d ≥ |I|+1 = 38 [i]
- linear OA(8167, 6561, F81, 34) (dual of [6561, 6494, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(813, 11, F81, 3) (dual of [11, 8, 4]-code or 11-arc in PG(2,81) or 11-cap in PG(2,81)), using
- discarding factors / shortening the dual code based on linear OA(813, 81, F81, 3) (dual of [81, 78, 4]-code or 81-arc in PG(2,81) or 81-cap in PG(2,81)), using
- Reed–Solomon code RS(78,81) [i]
- discarding factors / shortening the dual code based on linear OA(813, 81, F81, 3) (dual of [81, 78, 4]-code or 81-arc in PG(2,81) or 81-cap in PG(2,81)), using
- construction X applied to Ce(37) ⊂ Ce(33) [i] based on
- OOA 2-folding [i] based on linear OA(8178, 6572, F81, 38) (dual of [6572, 6494, 39]-code), using
- discarding factors / shortening the dual code based on linear OOA(8178, 3286, F81, 2, 38) (dual of [(3286, 2), 6494, 39]-NRT-code), using
(40, 40+38, 6775762)-Net in Base 81 — Upper bound on s
There is no (40, 78, 6775763)-net in base 81, because
- the generalized Rao bound for nets shows that 81m ≥ 72749 885518 999261 644433 844022 579760 019077 795360 847342 649481 695477 862672 339617 883866 086812 144912 843062 019013 920529 771009 075649 546795 542933 322144 190161 > 8178 [i]