Best Known (121−18, 121, s)-Nets in Base 5
(121−18, 121, 43407)-Net over F5 — Constructive and digital
Digital (103, 121, 43407)-net over F5, using
- 51 times duplication [i] based on digital (102, 120, 43407)-net over F5, using
- net defined by OOA [i] based on linear OOA(5120, 43407, F5, 18, 18) (dual of [(43407, 18), 781206, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(5120, 390663, F5, 18) (dual of [390663, 390543, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(5120, 390664, F5, 18) (dual of [390664, 390544, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(12) [i] based on
- linear OA(5113, 390625, F5, 18) (dual of [390625, 390512, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(581, 390625, F5, 13) (dual of [390625, 390544, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(57, 39, F5, 4) (dual of [39, 32, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(57, 44, F5, 4) (dual of [44, 37, 5]-code), using
- construction X applied to Ce(17) ⊂ Ce(12) [i] based on
- discarding factors / shortening the dual code based on linear OA(5120, 390664, F5, 18) (dual of [390664, 390544, 19]-code), using
- OA 9-folding and stacking [i] based on linear OA(5120, 390663, F5, 18) (dual of [390663, 390543, 19]-code), using
- net defined by OOA [i] based on linear OOA(5120, 43407, F5, 18, 18) (dual of [(43407, 18), 781206, 19]-NRT-code), using
(121−18, 121, 296987)-Net over F5 — Digital
Digital (103, 121, 296987)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5121, 296987, F5, 18) (dual of [296987, 296866, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(5121, 390665, F5, 18) (dual of [390665, 390544, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(12) [i] based on
- linear OA(5113, 390625, F5, 18) (dual of [390625, 390512, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(581, 390625, F5, 13) (dual of [390625, 390544, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(58, 40, F5, 4) (dual of [40, 32, 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(17) ⊂ Ce(12) [i] based on
- discarding factors / shortening the dual code based on linear OA(5121, 390665, F5, 18) (dual of [390665, 390544, 19]-code), using
(121−18, 121, large)-Net in Base 5 — Upper bound on s
There is no (103, 121, large)-net in base 5, because
- 16 times m-reduction [i] would yield (103, 105, large)-net in base 5, but