Best Known (102, 102+23, s)-Nets in Base 3
(102, 102+23, 688)-Net over F3 — Constructive and digital
Digital (102, 125, 688)-net over F3, using
- 31 times duplication [i] based on digital (101, 124, 688)-net over F3, using
- t-expansion [i] based on digital (100, 124, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 31, 172)-net over F81, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 7 and N(F) ≥ 172, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- trace code for nets [i] based on digital (7, 31, 172)-net over F81, using
- t-expansion [i] based on digital (100, 124, 688)-net over F3, using
(102, 102+23, 3290)-Net over F3 — Digital
Digital (102, 125, 3290)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3125, 3290, F3, 2, 23) (dual of [(3290, 2), 6455, 24]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3125, 6580, F3, 23) (dual of [6580, 6455, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(3125, 6581, F3, 23) (dual of [6581, 6456, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(19) [i] based on
- linear OA(3121, 6561, F3, 23) (dual of [6561, 6440, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(3105, 6561, F3, 20) (dual of [6561, 6456, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [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(22) ⊂ Ce(19) [i] based on
- discarding factors / shortening the dual code based on linear OA(3125, 6581, F3, 23) (dual of [6581, 6456, 24]-code), using
- OOA 2-folding [i] based on linear OA(3125, 6580, F3, 23) (dual of [6580, 6455, 24]-code), using
(102, 102+23, 586724)-Net in Base 3 — Upper bound on s
There is no (102, 125, 586725)-net in base 3, because
- 1 times m-reduction [i] would yield (102, 124, 586725)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 145559 399785 061026 364202 621654 060685 369313 884615 326208 671371 > 3124 [i]