Best Known (108, 108+23, s)-Nets in Base 3
(108, 108+23, 692)-Net over F3 — Constructive and digital
Digital (108, 131, 692)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (0, 11, 4)-net over F3, using
- net from sequence [i] based on digital (0, 3)-sequence over F3, using
- Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 0 and N(F) ≥ 4, using
- the rational function field F3(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 3)-sequence over F3, using
- 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
- digital (0, 11, 4)-net over F3, using
(108, 108+23, 3880)-Net over F3 — Digital
Digital (108, 131, 3880)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3131, 3880, F3, 23) (dual of [3880, 3749, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(3131, 6597, F3, 23) (dual of [6597, 6466, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(16) [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(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(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 Ce(22) ⊂ Ce(16) [i] based on
- discarding factors / shortening the dual code based on linear OA(3131, 6597, F3, 23) (dual of [6597, 6466, 24]-code), using
(108, 108+23, 1068281)-Net in Base 3 — Upper bound on s
There is no (108, 131, 1068282)-net in base 3, because
- 1 times m-reduction [i] would yield (108, 130, 1068282)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 106 112417 193777 595878 730659 512315 688580 469375 845709 705650 483105 > 3130 [i]