Best Known (102−19, 102, s)-Nets in Base 3
(102−19, 102, 731)-Net over F3 — Constructive and digital
Digital (83, 102, 731)-net over F3, using
- 31 times duplication [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
(102−19, 102, 3247)-Net over F3 — Digital
Digital (83, 102, 3247)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3102, 3247, F3, 2, 19) (dual of [(3247, 2), 6392, 20]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3102, 3291, F3, 2, 19) (dual of [(3291, 2), 6480, 20]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3102, 6582, F3, 19) (dual of [6582, 6480, 20]-code), using
- construction X applied to C([0,9]) ⊂ C([0,7]) [i] based on
- linear OA(397, 6562, F3, 19) (dual of [6562, 6465, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 316−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(381, 6562, F3, 15) (dual of [6562, 6481, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 316−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(35, 20, F3, 3) (dual of [20, 15, 4]-code or 20-cap in PG(4,3)), using
- construction X applied to C([0,9]) ⊂ C([0,7]) [i] based on
- OOA 2-folding [i] based on linear OA(3102, 6582, F3, 19) (dual of [6582, 6480, 20]-code), using
- discarding factors / shortening the dual code based on linear OOA(3102, 3291, F3, 2, 19) (dual of [(3291, 2), 6480, 20]-NRT-code), using
(102−19, 102, 468893)-Net in Base 3 — Upper bound on s
There is no (83, 102, 468894)-net in base 3, because
- 1 times m-reduction [i] would yield (83, 101, 468894)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 1 546143 722922 196923 958552 834505 952140 136161 843117 > 3101 [i]