Best Known (97, 97+22, s)-Nets in Base 3
(97, 97+22, 688)-Net over F3 — Constructive and digital
Digital (97, 119, 688)-net over F3, using
- 1 times m-reduction [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
(97, 97+22, 3292)-Net over F3 — Digital
Digital (97, 119, 3292)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3119, 3292, F3, 2, 22) (dual of [(3292, 2), 6465, 23]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3119, 6584, F3, 22) (dual of [6584, 6465, 23]-code), using
- construction XX applied to Ce(21) ⊂ Ce(18) ⊂ Ce(16) [i] based on
- 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(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(389, 6561, F3, 17) (dual of [6561, 6472, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(34, 21, F3, 2) (dual of [21, 17, 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
- 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(21) ⊂ Ce(18) ⊂ Ce(16) [i] based on
- OOA 2-folding [i] based on linear OA(3119, 6584, F3, 22) (dual of [6584, 6465, 23]-code), using
(97, 97+22, 356086)-Net in Base 3 — Upper bound on s
There is no (97, 119, 356087)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 599 007698 777616 188236 049716 680057 168899 890964 289429 673035 > 3119 [i]