Best Known (156−23, 156, s)-Nets in Base 3
(156−23, 156, 5370)-Net over F3 — Constructive and digital
Digital (133, 156, 5370)-net over F3, using
- 31 times duplication [i] based on digital (132, 155, 5370)-net over F3, using
- net defined by OOA [i] based on linear OOA(3155, 5370, F3, 23, 23) (dual of [(5370, 23), 123355, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(3155, 59071, F3, 23) (dual of [59071, 58916, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(3155, 59073, F3, 23) (dual of [59073, 58918, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(19) [i] based on
- linear OA(3151, 59049, F3, 23) (dual of [59049, 58898, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- 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(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(22) ⊂ Ce(19) [i] based on
- discarding factors / shortening the dual code based on linear OA(3155, 59073, F3, 23) (dual of [59073, 58918, 24]-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(3155, 59071, F3, 23) (dual of [59071, 58916, 24]-code), using
- net defined by OOA [i] based on linear OOA(3155, 5370, F3, 23, 23) (dual of [(5370, 23), 123355, 24]-NRT-code), using
(156−23, 156, 19691)-Net over F3 — Digital
Digital (133, 156, 19691)-net over F3, using
- 31 times duplication [i] based on digital (132, 155, 19691)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3155, 19691, F3, 3, 23) (dual of [(19691, 3), 58918, 24]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3155, 59073, F3, 23) (dual of [59073, 58918, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(19) [i] based on
- linear OA(3151, 59049, F3, 23) (dual of [59049, 58898, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- 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(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(22) ⊂ Ce(19) [i] based on
- OOA 3-folding [i] based on linear OA(3155, 59073, F3, 23) (dual of [59073, 58918, 24]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3155, 19691, F3, 3, 23) (dual of [(19691, 3), 58918, 24]-NRT-code), using
(156−23, 156, large)-Net in Base 3 — Upper bound on s
There is no (133, 156, large)-net in base 3, because
- 21 times m-reduction [i] would yield (133, 135, large)-net in base 3, but