Best Known (18, 18+22, s)-Nets in Base 81
(18, 18+22, 370)-Net over F81 — Constructive and digital
Digital (18, 40, 370)-net over F81, using
- t-expansion [i] based on digital (16, 40, 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
(18, 18+22, 480)-Net over F81 — Digital
Digital (18, 40, 480)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8140, 480, F81, 22) (dual of [480, 440, 23]-code), using
- 67 step Varšamov–Edel lengthening with (ri) = (1, 66 times 0) [i] based on linear OA(8139, 412, F81, 22) (dual of [412, 373, 23]-code), using
- construction X applied to C([30,51]) ⊂ C([31,51]) [i] based on
- linear OA(8139, 410, F81, 22) (dual of [410, 371, 23]-code), using the BCH-code C(I) with length 410 | 812−1, defining interval I = {30,31,…,51}, and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(8137, 410, F81, 21) (dual of [410, 373, 22]-code), using the BCH-code C(I) with length 410 | 812−1, defining interval I = {31,32,…,51}, and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(810, 2, F81, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(810, s, F81, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to C([30,51]) ⊂ C([31,51]) [i] based on
- 67 step Varšamov–Edel lengthening with (ri) = (1, 66 times 0) [i] based on linear OA(8139, 412, F81, 22) (dual of [412, 373, 23]-code), using
(18, 18+22, 534398)-Net in Base 81 — Upper bound on s
There is no (18, 40, 534399)-net in base 81, because
- the generalized Rao bound for nets shows that 81m ≥ 21847 516184 051300 601108 027692 354717 193797 831572 731825 911410 588475 343471 871121 > 8140 [i]