Best Known (99, 99+31, s)-Nets in Base 3
(99, 99+31, 400)-Net over F3 — Constructive and digital
Digital (99, 130, 400)-net over F3, using
- 32 times duplication [i] based on digital (97, 128, 400)-net over F3, using
- trace code for nets [i] based on digital (1, 32, 100)-net over F81, using
- net from sequence [i] based on digital (1, 99)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 1 and N(F) ≥ 100, using
- net from sequence [i] based on digital (1, 99)-sequence over F81, using
- trace code for nets [i] based on digital (1, 32, 100)-net over F81, using
(99, 99+31, 748)-Net over F3 — Digital
Digital (99, 130, 748)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3130, 748, F3, 31) (dual of [748, 618, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(3130, 763, F3, 31) (dual of [763, 633, 32]-code), using
- construction XX applied to Ce(30) ⊂ Ce(24) ⊂ Ce(22) [i] based on
- linear OA(3118, 729, F3, 31) (dual of [729, 611, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(397, 729, F3, 25) (dual of [729, 632, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(391, 729, F3, 23) (dual of [729, 638, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(310, 32, F3, 5) (dual of [32, 22, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(310, 36, F3, 5) (dual of [36, 26, 6]-code), using
- (u, u−v, u+v+w)-construction [i] based on
- linear OA(31, 12, F3, 1) (dual of [12, 11, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(31, s, F3, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- linear OA(33, 12, F3, 2) (dual of [12, 9, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(33, 13, F3, 2) (dual of [13, 10, 3]-code), using
- Hamming code H(3,3) [i]
- discarding factors / shortening the dual code based on linear OA(33, 13, F3, 2) (dual of [13, 10, 3]-code), using
- linear OA(36, 12, F3, 5) (dual of [12, 6, 6]-code), using
- extended Golay code Ge(3) [i]
- linear OA(31, 12, F3, 1) (dual of [12, 11, 2]-code), using
- (u, u−v, u+v+w)-construction [i] based on
- discarding factors / shortening the dual code based on linear OA(310, 36, F3, 5) (dual of [36, 26, 6]-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(30) ⊂ Ce(24) ⊂ Ce(22) [i] based on
- discarding factors / shortening the dual code based on linear OA(3130, 763, F3, 31) (dual of [763, 633, 32]-code), using
(99, 99+31, 40721)-Net in Base 3 — Upper bound on s
There is no (99, 130, 40722)-net in base 3, because
- 1 times m-reduction [i] would yield (99, 129, 40722)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 35 375750 823667 317809 973802 942032 501787 721577 781868 907331 217241 > 3129 [i]