Best Known (157, 157+31, s)-Nets in Base 3
(157, 157+31, 1480)-Net over F3 — Constructive and digital
Digital (157, 188, 1480)-net over F3, using
- trace code for nets [i] based on digital (16, 47, 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
(157, 157+31, 8318)-Net over F3 — Digital
Digital (157, 188, 8318)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3188, 8318, F3, 2, 31) (dual of [(8318, 2), 16448, 32]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3188, 9855, F3, 2, 31) (dual of [(9855, 2), 19522, 32]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3188, 19710, F3, 31) (dual of [19710, 19522, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(25) [i] based on
- linear OA(3181, 19683, F3, 31) (dual of [19683, 19502, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(3154, 19683, F3, 26) (dual of [19683, 19529, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(37, 27, F3, 4) (dual of [27, 20, 5]-code), using
- an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 26 = 33−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- construction X applied to Ce(30) ⊂ Ce(25) [i] based on
- OOA 2-folding [i] based on linear OA(3188, 19710, F3, 31) (dual of [19710, 19522, 32]-code), using
- discarding factors / shortening the dual code based on linear OOA(3188, 9855, F3, 2, 31) (dual of [(9855, 2), 19522, 32]-NRT-code), using
(157, 157+31, 2850007)-Net in Base 3 — Upper bound on s
There is no (157, 188, 2850008)-net in base 3, because
- 1 times m-reduction [i] would yield (157, 187, 2850008)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 166599 868125 921332 477302 519287 420924 722447 011531 975221 800583 556314 601525 978230 753342 532641 > 3187 [i]