Best Known (102−18, 102, s)-Nets in Base 3
(102−18, 102, 731)-Net over F3 — Constructive and digital
Digital (84, 102, 731)-net over F3, using
- 31 times duplication [i] based on digital (83, 101, 731)-net over F3, using
- t-expansion [i] based on digital (82, 101, 731)-net over F3, using
- net defined by OOA [i] based on linear OOA(3101, 731, F3, 19, 19) (dual of [(731, 19), 13788, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(3101, 6580, F3, 19) (dual of [6580, 6479, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(3101, 6581, F3, 19) (dual of [6581, 6480, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(15) [i] based on
- linear OA(397, 6561, F3, 19) (dual of [6561, 6464, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(381, 6561, F3, 16) (dual of [6561, 6480, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(34, 20, F3, 2) (dual of [20, 16, 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(3101, 6581, F3, 19) (dual of [6581, 6480, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(3101, 6580, F3, 19) (dual of [6580, 6479, 20]-code), using
- net defined by OOA [i] based on linear OOA(3101, 731, F3, 19, 19) (dual of [(731, 19), 13788, 20]-NRT-code), using
- t-expansion [i] based on digital (82, 101, 731)-net over F3, using
(102−18, 102, 3479)-Net over F3 — Digital
Digital (84, 102, 3479)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3102, 3479, F3, 18) (dual of [3479, 3377, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(3102, 6585, F3, 18) (dual of [6585, 6483, 19]-code), using
- construction XX applied to Ce(18) ⊂ Ce(15) ⊂ Ce(13) [i] based on
- linear OA(397, 6561, F3, 19) (dual of [6561, 6464, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(381, 6561, F3, 16) (dual of [6561, 6480, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(373, 6561, F3, 14) (dual of [6561, 6488, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [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
- linear OA(31, 4, F3, 1) (dual of [4, 3, 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 (see above)
- construction XX applied to Ce(18) ⊂ Ce(15) ⊂ Ce(13) [i] based on
- discarding factors / shortening the dual code based on linear OA(3102, 6585, F3, 18) (dual of [6585, 6483, 19]-code), using
(102−18, 102, 529771)-Net in Base 3 — Upper bound on s
There is no (84, 102, 529772)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 4 638426 162006 057357 271938 958493 110903 344764 351577 > 3102 [i]