Best Known (135−20, 135, s)-Nets in Base 3
(135−20, 135, 5907)-Net over F3 — Constructive and digital
Digital (115, 135, 5907)-net over F3, using
- net defined by OOA [i] based on linear OOA(3135, 5907, F3, 20, 20) (dual of [(5907, 20), 118005, 21]-NRT-code), using
- OA 10-folding and stacking [i] based on linear OA(3135, 59070, F3, 20) (dual of [59070, 58935, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(3135, 59073, F3, 20) (dual of [59073, 58938, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(16) [i] based on
- linear OA(3131, 59049, F3, 20) (dual of [59049, 58918, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(3111, 59049, F3, 17) (dual of [59049, 58938, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(34, 24, F3, 2) (dual of [24, 20, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- Hamming code H(4,3) [i]
- discarding factors / shortening the dual code based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- construction X applied to Ce(19) ⊂ Ce(16) [i] based on
- discarding factors / shortening the dual code based on linear OA(3135, 59073, F3, 20) (dual of [59073, 58938, 21]-code), using
- OA 10-folding and stacking [i] based on linear OA(3135, 59070, F3, 20) (dual of [59070, 58935, 21]-code), using
(135−20, 135, 19691)-Net over F3 — Digital
Digital (115, 135, 19691)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3135, 19691, F3, 3, 20) (dual of [(19691, 3), 58938, 21]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3135, 59073, F3, 20) (dual of [59073, 58938, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(16) [i] based on
- linear OA(3131, 59049, F3, 20) (dual of [59049, 58918, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(3111, 59049, F3, 17) (dual of [59049, 58938, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(34, 24, F3, 2) (dual of [24, 20, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- Hamming code H(4,3) [i]
- discarding factors / shortening the dual code based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- construction X applied to Ce(19) ⊂ Ce(16) [i] based on
- OOA 3-folding [i] based on linear OA(3135, 59073, F3, 20) (dual of [59073, 58938, 21]-code), using
(135−20, 135, 6252915)-Net in Base 3 — Upper bound on s
There is no (115, 135, 6252916)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 25785 138423 523673 811579 614542 386393 519939 242031 853968 497624 904537 > 3135 [i]