Best Known (148−36, 148, s)-Nets in Base 5
(148−36, 148, 504)-Net over F5 — Constructive and digital
Digital (112, 148, 504)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (38, 56, 252)-net over F5, using
- trace code for nets [i] based on digital (10, 28, 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, 28, 126)-net over F25, using
- digital (56, 92, 252)-net over F5, using
- trace code for nets [i] based on digital (10, 46, 126)-net over F25, using
- net from sequence [i] based on digital (10, 125)-sequence over F25 (see above)
- trace code for nets [i] based on digital (10, 46, 126)-net over F25, using
- digital (38, 56, 252)-net over F5, using
(148−36, 148, 3248)-Net over F5 — Digital
Digital (112, 148, 3248)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5148, 3248, F5, 36) (dual of [3248, 3100, 37]-code), using
- 111 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 5 times 0, 1, 14 times 0, 1, 29 times 0, 1, 57 times 0) [i] based on linear OA(5142, 3131, F5, 36) (dual of [3131, 2989, 37]-code), using
- construction X applied to Ce(35) ⊂ Ce(33) [i] based on
- linear OA(5141, 3125, F5, 36) (dual of [3125, 2984, 37]-code), using an extension Ce(35) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,35], and designed minimum distance d ≥ |I|+1 = 36 [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(51, 6, F5, 1) (dual of [6, 5, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(51, s, F5, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(35) ⊂ Ce(33) [i] based on
- 111 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 5 times 0, 1, 14 times 0, 1, 29 times 0, 1, 57 times 0) [i] based on linear OA(5142, 3131, F5, 36) (dual of [3131, 2989, 37]-code), using
(148−36, 148, 1054754)-Net in Base 5 — Upper bound on s
There is no (112, 148, 1054755)-net in base 5, because
- 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]