Best Known (200−33, 200, s)-Nets in Base 3
(200−33, 200, 1480)-Net over F3 — Constructive and digital
Digital (167, 200, 1480)-net over F3, using
- t-expansion [i] based on digital (166, 200, 1480)-net over F3, using
- trace code for nets [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
- trace code for nets [i] based on digital (16, 50, 370)-net over F81, using
(200−33, 200, 8458)-Net over F3 — Digital
Digital (167, 200, 8458)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3200, 8458, F3, 2, 33) (dual of [(8458, 2), 16716, 34]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3200, 9851, F3, 2, 33) (dual of [(9851, 2), 19502, 34]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3200, 19702, F3, 33) (dual of [19702, 19502, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(3200, 19703, F3, 33) (dual of [19703, 19503, 34]-code), using
- construction X applied to C([0,16]) ⊂ C([0,15]) [i] based on
- linear OA(3199, 19684, F3, 33) (dual of [19684, 19485, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684 | 318−1, defining interval I = [0,16], and minimum distance d ≥ |{−16,−15,…,16}|+1 = 34 (BCH-bound) [i]
- linear OA(3181, 19684, F3, 31) (dual of [19684, 19503, 32]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684 | 318−1, defining interval I = [0,15], and minimum distance d ≥ |{−15,−14,…,15}|+1 = 32 (BCH-bound) [i]
- linear OA(31, 19, F3, 1) (dual of [19, 18, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(31, s, F3, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,16]) ⊂ C([0,15]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3200, 19703, F3, 33) (dual of [19703, 19503, 34]-code), using
- OOA 2-folding [i] based on linear OA(3200, 19702, F3, 33) (dual of [19702, 19502, 34]-code), using
- discarding factors / shortening the dual code based on linear OOA(3200, 9851, F3, 2, 33) (dual of [(9851, 2), 19502, 34]-NRT-code), using
(200−33, 200, 2922146)-Net in Base 3 — Upper bound on s
There is no (167, 200, 2922147)-net in base 3, because
- 1 times m-reduction [i] would yield (167, 199, 2922147)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 88538 089715 994225 676333 406078 173872 027457 304664 736432 814084 907214 420318 483570 667208 183674 685665 > 3199 [i]