Best Known (122−23, 122, s)-Nets in Base 3
(122−23, 122, 688)-Net over F3 — Constructive and digital
Digital (99, 122, 688)-net over F3, using
- 32 times duplication [i] based on digital (97, 120, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 30, 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, 30, 172)-net over F81, using
(122−23, 122, 3008)-Net over F3 — Digital
Digital (99, 122, 3008)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3122, 3008, F3, 2, 23) (dual of [(3008, 2), 5894, 24]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3122, 3285, F3, 2, 23) (dual of [(3285, 2), 6448, 24]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3122, 6570, F3, 23) (dual of [6570, 6448, 24]-code), using
- 1 times code embedding in larger space [i] based on linear OA(3121, 6569, F3, 23) (dual of [6569, 6448, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(21) [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(30, 8, F3, 0) (dual of [8, 8, 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
- construction X applied to Ce(22) ⊂ Ce(21) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(3121, 6569, F3, 23) (dual of [6569, 6448, 24]-code), using
- OOA 2-folding [i] based on linear OA(3122, 6570, F3, 23) (dual of [6570, 6448, 24]-code), using
- discarding factors / shortening the dual code based on linear OOA(3122, 3285, F3, 2, 23) (dual of [(3285, 2), 6448, 24]-NRT-code), using
(122−23, 122, 434817)-Net in Base 3 — Upper bound on s
There is no (99, 122, 434818)-net in base 3, because
- 1 times m-reduction [i] would yield (99, 121, 434818)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 5391 036198 074366 700700 404401 212063 016124 853914 233725 854849 > 3121 [i]