Best Known (150−37, 150, s)-Nets in Base 5
(150−37, 150, 504)-Net over F5 — Constructive and digital
Digital (113, 150, 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 (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 (see above)
- trace code for nets [i] based on digital (10, 47, 126)-net over F25, using
- digital (38, 56, 252)-net over F5, using
(150−37, 150, 3164)-Net over F5 — Digital
Digital (113, 150, 3164)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5150, 3164, F5, 37) (dual of [3164, 3014, 38]-code), using
- 30 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 1, 7 times 0, 1, 18 times 0) [i] based on linear OA(5146, 3130, F5, 37) (dual of [3130, 2984, 38]-code), using
- construction X applied to Ce(36) ⊂ Ce(35) [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(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(50, 5, F5, 0) (dual of [5, 5, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(50, s, F5, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(36) ⊂ Ce(35) [i] based on
- 30 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 1, 7 times 0, 1, 18 times 0) [i] based on linear OA(5146, 3130, F5, 37) (dual of [3130, 2984, 38]-code), using
(150−37, 150, 1153409)-Net in Base 5 — Upper bound on s
There is no (113, 150, 1153410)-net in base 5, because
- 1 times m-reduction [i] would yield (113, 149, 1153410)-net in base 5, but
- the generalized Rao bound for nets shows that 5m ≥ 140 130849 312875 375797 866307 418314 139210 517808 667254 008057 858114 231572 532207 807306 798648 896601 036965 239105 > 5149 [i]