Best Known (98, 98+25, s)-Nets in Base 3
(98, 98+25, 640)-Net over F3 — Constructive and digital
Digital (98, 123, 640)-net over F3, using
- 1 times m-reduction [i] based on digital (98, 124, 640)-net over F3, using
- trace code for nets [i] based on digital (5, 31, 160)-net over F81, using
- net from sequence [i] based on digital (5, 159)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 5 and N(F) ≥ 160, using
- net from sequence [i] based on digital (5, 159)-sequence over F81, using
- trace code for nets [i] based on digital (5, 31, 160)-net over F81, using
(98, 98+25, 1579)-Net over F3 — Digital
Digital (98, 123, 1579)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3123, 1579, F3, 25) (dual of [1579, 1456, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(3123, 2224, F3, 25) (dual of [2224, 2101, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,9]) [i] based on
- linear OA(3113, 2188, F3, 25) (dual of [2188, 2075, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 2188 | 314−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(385, 2188, F3, 19) (dual of [2188, 2103, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 2188 | 314−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- 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
- construction X applied to C([0,12]) ⊂ C([0,9]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3123, 2224, F3, 25) (dual of [2224, 2101, 26]-code), using
(98, 98+25, 187515)-Net in Base 3 — Upper bound on s
There is no (98, 123, 187516)-net in base 3, because
- 1 times m-reduction [i] would yield (98, 122, 187516)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 16173 445248 701869 035627 359384 518786 764055 568168 682455 043393 > 3122 [i]