Best Known (123−23, 123, s)-Nets in Base 3
(123−23, 123, 688)-Net over F3 — Constructive and digital
Digital (100, 123, 688)-net over F3, using
- 1 times m-reduction [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
(123−23, 123, 3179)-Net over F3 — Digital
Digital (100, 123, 3179)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3123, 3179, F3, 2, 23) (dual of [(3179, 2), 6235, 24]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3123, 3286, F3, 2, 23) (dual of [(3286, 2), 6449, 24]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3123, 6572, F3, 23) (dual of [6572, 6449, 24]-code), using
- construction XX applied to Ce(22) ⊂ Ce(21) ⊂ 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(3113, 6561, F3, 22) (dual of [6561, 6448, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [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(30, 9, F3, 0) (dual of [9, 9, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(30, s, F3, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(31, 2, F3, 1) (dual of [2, 1, 2]-code), using
- dual of repetition code with length 2 [i]
- construction XX applied to Ce(22) ⊂ Ce(21) ⊂ Ce(19) [i] based on
- OOA 2-folding [i] based on linear OA(3123, 6572, F3, 23) (dual of [6572, 6449, 24]-code), using
- discarding factors / shortening the dual code based on linear OOA(3123, 3286, F3, 2, 23) (dual of [(3286, 2), 6449, 24]-NRT-code), using
(123−23, 123, 480488)-Net in Base 3 — Upper bound on s
There is no (100, 123, 480489)-net in base 3, because
- 1 times m-reduction [i] would yield (100, 122, 480489)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 16173 222981 682498 151069 282373 905637 475574 432673 987887 375323 > 3122 [i]