Best Known (113−18, 113, s)-Nets in Base 3
(113−18, 113, 2189)-Net over F3 — Constructive and digital
Digital (95, 113, 2189)-net over F3, using
- t-expansion [i] based on digital (94, 113, 2189)-net over F3, using
- net defined by OOA [i] based on linear OOA(3113, 2189, F3, 19, 19) (dual of [(2189, 19), 41478, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(3113, 19702, F3, 19) (dual of [19702, 19589, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(3113, 19705, F3, 19) (dual of [19705, 19592, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(15) [i] based on
- linear OA(3109, 19683, F3, 19) (dual of [19683, 19574, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(391, 19683, F3, 16) (dual of [19683, 19592, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(34, 22, F3, 2) (dual of [22, 18, 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(18) ⊂ Ce(15) [i] based on
- discarding factors / shortening the dual code based on linear OA(3113, 19705, F3, 19) (dual of [19705, 19592, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(3113, 19702, F3, 19) (dual of [19702, 19589, 20]-code), using
- net defined by OOA [i] based on linear OOA(3113, 2189, F3, 19, 19) (dual of [(2189, 19), 41478, 20]-NRT-code), using
(113−18, 113, 9853)-Net over F3 — Digital
Digital (95, 113, 9853)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3113, 9853, F3, 2, 18) (dual of [(9853, 2), 19593, 19]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(3111, 9852, F3, 2, 18) (dual of [(9852, 2), 19593, 19]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3111, 19704, F3, 18) (dual of [19704, 19593, 19]-code), using
- 1 times code embedding in larger space [i] based on linear OA(3110, 19703, F3, 18) (dual of [19703, 19593, 19]-code), using
- construction X4 applied to Ce(18) ⊂ Ce(15) [i] based on
- linear OA(3109, 19683, F3, 19) (dual of [19683, 19574, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(391, 19683, F3, 16) (dual of [19683, 19592, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(319, 20, F3, 19) (dual of [20, 1, 20]-code or 20-arc in PG(18,3)), using
- dual of repetition code with length 20 [i]
- linear OA(31, 20, F3, 1) (dual of [20, 19, 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 X4 applied to Ce(18) ⊂ Ce(15) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(3110, 19703, F3, 18) (dual of [19703, 19593, 19]-code), using
- OOA 2-folding [i] based on linear OA(3111, 19704, F3, 18) (dual of [19704, 19593, 19]-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(3111, 9852, F3, 2, 18) (dual of [(9852, 2), 19593, 19]-NRT-code), using
(113−18, 113, 2028813)-Net in Base 3 — Upper bound on s
There is no (95, 113, 2028814)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 821681 297028 588146 048101 148077 746748 966939 524218 933901 > 3113 [i]