Best Known (112, 112+37, s)-Nets in Base 5
(112, 112+37, 460)-Net over F5 — Constructive and digital
Digital (112, 149, 460)-net over F5, using
- 51 times duplication [i] based on digital (111, 148, 460)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (36, 54, 208)-net over F5, using
- trace code for nets [i] based on digital (9, 27, 104)-net over F25, using
- net from sequence [i] based on digital (9, 103)-sequence over F25, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 9 and N(F) ≥ 104, using
- net from sequence [i] based on digital (9, 103)-sequence over F25, using
- trace code for nets [i] based on digital (9, 27, 104)-net over F25, using
- digital (57, 94, 252)-net over F5, using
- trace code for nets [i] based on digital (10, 47, 126)-net over F25, using
- net from sequence [i] based on digital (10, 125)-sequence over F25, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 10 and N(F) ≥ 126, using
- the Hermitian function field over F25 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 10 and N(F) ≥ 126, using
- net from sequence [i] based on digital (10, 125)-sequence over F25, using
- trace code for nets [i] based on digital (10, 47, 126)-net over F25, using
- digital (36, 54, 208)-net over F5, using
- (u, u+v)-construction [i] based on
(112, 112+37, 3115)-Net over F5 — Digital
Digital (112, 149, 3115)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5149, 3115, F5, 37) (dual of [3115, 2966, 38]-code), using
- discarding factors / shortening the dual code based on linear OA(5149, 3138, F5, 37) (dual of [3138, 2989, 38]-code), using
- construction X applied to Ce(36) ⊂ Ce(33) [i] based on
- linear OA(5146, 3125, F5, 37) (dual of [3125, 2979, 38]-code), using an extension Ce(36) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,36], and designed minimum distance d ≥ |I|+1 = 37 [i]
- linear OA(5136, 3125, F5, 34) (dual of [3125, 2989, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(53, 13, F5, 2) (dual of [13, 10, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(53, 24, F5, 2) (dual of [24, 21, 3]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 24 = 52−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 3 [i]
- discarding factors / shortening the dual code based on linear OA(53, 24, F5, 2) (dual of [24, 21, 3]-code), using
- construction X applied to Ce(36) ⊂ Ce(33) [i] based on
- discarding factors / shortening the dual code based on linear OA(5149, 3138, F5, 37) (dual of [3138, 2989, 38]-code), using
(112, 112+37, 1054754)-Net in Base 5 — Upper bound on s
There is no (112, 149, 1054755)-net in base 5, because
- 1 times m-reduction [i] would yield (112, 148, 1054755)-net in base 5, but
- the generalized Rao bound for nets shows that 5m ≥ 28 026175 833631 021657 688099 821862 003977 668056 558935 065759 663645 937230 398134 072493 833379 586190 796877 895225 > 5148 [i]