Best Known (126, 126+19, s)-Nets in Base 5
(126, 126+19, 217020)-Net over F5 — Constructive and digital
Digital (126, 145, 217020)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (0, 9, 6)-net over F5, using
- net from sequence [i] based on digital (0, 5)-sequence over F5, using
- Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F5 with g(F) = 0 and N(F) ≥ 6, using
- the rational function field F5(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 5)-sequence over F5, using
- digital (117, 136, 217014)-net over F5, using
- net defined by OOA [i] based on linear OOA(5136, 217014, F5, 19, 19) (dual of [(217014, 19), 4123130, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(5136, 1953127, F5, 19) (dual of [1953127, 1952991, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(5136, 1953134, F5, 19) (dual of [1953134, 1952998, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(17) [i] based on
- linear OA(5136, 1953125, F5, 19) (dual of [1953125, 1952989, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(5127, 1953125, F5, 18) (dual of [1953125, 1952998, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(50, 9, F5, 0) (dual of [9, 9, 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(18) ⊂ Ce(17) [i] based on
- discarding factors / shortening the dual code based on linear OA(5136, 1953134, F5, 19) (dual of [1953134, 1952998, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(5136, 1953127, F5, 19) (dual of [1953127, 1952991, 20]-code), using
- net defined by OOA [i] based on linear OOA(5136, 217014, F5, 19, 19) (dual of [(217014, 19), 4123130, 20]-NRT-code), using
- digital (0, 9, 6)-net over F5, using
(126, 126+19, 1494751)-Net over F5 — Digital
Digital (126, 145, 1494751)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5145, 1494751, F5, 19) (dual of [1494751, 1494606, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(5145, 1953135, F5, 19) (dual of [1953135, 1952990, 20]-code), using
- (u, u+v)-construction [i] based on
- linear OA(59, 10, F5, 9) (dual of [10, 1, 10]-code or 10-arc in PG(8,5)), using
- dual of repetition code with length 10 [i]
- linear OA(5136, 1953125, F5, 19) (dual of [1953125, 1952989, 20]-code), using
- an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(59, 10, F5, 9) (dual of [10, 1, 10]-code or 10-arc in PG(8,5)), using
- (u, u+v)-construction [i] based on
- discarding factors / shortening the dual code based on linear OA(5145, 1953135, F5, 19) (dual of [1953135, 1952990, 20]-code), using
(126, 126+19, large)-Net in Base 5 — Upper bound on s
There is no (126, 145, large)-net in base 5, because
- 17 times m-reduction [i] would yield (126, 128, large)-net in base 5, but