Best Known (135−18, 135, s)-Nets in Base 5
(135−18, 135, 217018)-Net over F5 — Constructive and digital
Digital (117, 135, 217018)-net over F5, using
- 51 times duplication [i] based on digital (116, 134, 217018)-net over F5, using
- net defined by OOA [i] based on linear OOA(5134, 217018, F5, 18, 18) (dual of [(217018, 18), 3906190, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(5134, 1953162, F5, 18) (dual of [1953162, 1953028, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(5134, 1953168, F5, 18) (dual of [1953168, 1953034, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(12) [i] based on
- 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(591, 1953125, F5, 13) (dual of [1953125, 1953034, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(57, 43, F5, 4) (dual of [43, 36, 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(5134, 1953168, F5, 18) (dual of [1953168, 1953034, 19]-code), using
- OA 9-folding and stacking [i] based on linear OA(5134, 1953162, F5, 18) (dual of [1953162, 1953028, 19]-code), using
- net defined by OOA [i] based on linear OOA(5134, 217018, F5, 18, 18) (dual of [(217018, 18), 3906190, 19]-NRT-code), using
(135−18, 135, 1214363)-Net over F5 — Digital
Digital (117, 135, 1214363)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5135, 1214363, F5, 18) (dual of [1214363, 1214228, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(5135, 1953169, F5, 18) (dual of [1953169, 1953034, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(12) [i] based on
- 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(591, 1953125, F5, 13) (dual of [1953125, 1953034, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(58, 44, F5, 4) (dual of [44, 36, 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(5135, 1953169, F5, 18) (dual of [1953169, 1953034, 19]-code), using
(135−18, 135, large)-Net in Base 5 — Upper bound on s
There is no (117, 135, large)-net in base 5, because
- 16 times m-reduction [i] would yield (117, 119, large)-net in base 5, but