Best Known (128−22, 128, s)-Nets in Base 5
(128−22, 128, 7105)-Net over F5 — Constructive and digital
Digital (106, 128, 7105)-net over F5, using
- 52 times duplication [i] based on digital (104, 126, 7105)-net over F5, using
- net defined by OOA [i] based on linear OOA(5126, 7105, F5, 22, 22) (dual of [(7105, 22), 156184, 23]-NRT-code), using
- OA 11-folding and stacking [i] based on linear OA(5126, 78155, F5, 22) (dual of [78155, 78029, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(16) [i] based on
- linear OA(5120, 78125, F5, 22) (dual of [78125, 78005, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(592, 78125, F5, 17) (dual of [78125, 78033, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(56, 30, F5, 4) (dual of [30, 24, 5]-code), using
- construction X applied to Ce(21) ⊂ Ce(16) [i] based on
- OA 11-folding and stacking [i] based on linear OA(5126, 78155, F5, 22) (dual of [78155, 78029, 23]-code), using
- net defined by OOA [i] based on linear OOA(5126, 7105, F5, 22, 22) (dual of [(7105, 22), 156184, 23]-NRT-code), using
(128−22, 128, 56964)-Net over F5 — Digital
Digital (106, 128, 56964)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5128, 56964, F5, 22) (dual of [56964, 56836, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(5128, 78161, F5, 22) (dual of [78161, 78033, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(16) [i] based on
- linear OA(5120, 78125, F5, 22) (dual of [78125, 78005, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(592, 78125, F5, 17) (dual of [78125, 78033, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(58, 36, F5, 4) (dual of [36, 28, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(58, 52, F5, 4) (dual of [52, 44, 5]-code), using
- trace code [i] based on linear OA(254, 26, F25, 4) (dual of [26, 22, 5]-code or 26-arc in PG(3,25)), using
- extended Reed–Solomon code RSe(22,25) [i]
- algebraic-geometric code AG(F, Q+9P) with degQ = 3 and degPÂ =Â 2 [i] based on function field F/F25 with g(F) = 0 and N(F) ≥ 26, using the rational function field F25(x) [i]
- algebraic-geometric code AG(F,7P) with degPÂ =Â 3 [i] based on function field F/F25 with g(F) = 0 and N(F) ≥ 26 (see above)
- trace code [i] based on linear OA(254, 26, F25, 4) (dual of [26, 22, 5]-code or 26-arc in PG(3,25)), using
- discarding factors / shortening the dual code based on linear OA(58, 52, F5, 4) (dual of [52, 44, 5]-code), using
- construction X applied to Ce(21) ⊂ Ce(16) [i] based on
- discarding factors / shortening the dual code based on linear OA(5128, 78161, F5, 22) (dual of [78161, 78033, 23]-code), using
(128−22, 128, large)-Net in Base 5 — Upper bound on s
There is no (106, 128, large)-net in base 5, because
- 20 times m-reduction [i] would yield (106, 108, large)-net in base 5, but