Best Known (123−26, 123, s)-Nets in Base 3
(123−26, 123, 600)-Net over F3 — Constructive and digital
Digital (97, 123, 600)-net over F3, using
- 1 times m-reduction [i] based on digital (97, 124, 600)-net over F3, using
- trace code for nets [i] based on digital (4, 31, 150)-net over F81, using
- net from sequence [i] based on digital (4, 149)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 4 and N(F) ≥ 150, using
- net from sequence [i] based on digital (4, 149)-sequence over F81, using
- trace code for nets [i] based on digital (4, 31, 150)-net over F81, using
(123−26, 123, 1283)-Net over F3 — Digital
Digital (97, 123, 1283)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3123, 1283, F3, 26) (dual of [1283, 1160, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(3123, 2200, F3, 26) (dual of [2200, 2077, 27]-code), using
- construction X applied to Ce(25) ⊂ Ce(22) [i] based on
- linear OA(3120, 2187, F3, 26) (dual of [2187, 2067, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(3106, 2187, F3, 23) (dual of [2187, 2081, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(33, 13, F3, 2) (dual of [13, 10, 3]-code), using
- Hamming code H(3,3) [i]
- construction X applied to Ce(25) ⊂ Ce(22) [i] based on
- discarding factors / shortening the dual code based on linear OA(3123, 2200, F3, 26) (dual of [2200, 2077, 27]-code), using
(123−26, 123, 92601)-Net in Base 3 — Upper bound on s
There is no (97, 123, 92602)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 48521 744625 606484 735849 164976 070943 978528 790740 217975 550589 > 3123 [i]