Best Known (137−21, 137, s)-Nets in Base 5
(137−21, 137, 39066)-Net over F5 — Constructive and digital
Digital (116, 137, 39066)-net over F5, using
- 51 times duplication [i] based on digital (115, 136, 39066)-net over F5, using
- net defined by OOA [i] based on linear OOA(5136, 39066, F5, 21, 21) (dual of [(39066, 21), 820250, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(5136, 390661, F5, 21) (dual of [390661, 390525, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(5136, 390664, F5, 21) (dual of [390664, 390528, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(15) [i] based on
- linear OA(5129, 390625, F5, 21) (dual of [390625, 390496, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(597, 390625, F5, 16) (dual of [390625, 390528, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [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(20) ⊂ Ce(15) [i] based on
- discarding factors / shortening the dual code based on linear OA(5136, 390664, F5, 21) (dual of [390664, 390528, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(5136, 390661, F5, 21) (dual of [390661, 390525, 22]-code), using
- net defined by OOA [i] based on linear OOA(5136, 39066, F5, 21, 21) (dual of [(39066, 21), 820250, 22]-NRT-code), using
(137−21, 137, 199655)-Net over F5 — Digital
Digital (116, 137, 199655)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5137, 199655, F5, 21) (dual of [199655, 199518, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(5137, 390665, F5, 21) (dual of [390665, 390528, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(15) [i] based on
- linear OA(5129, 390625, F5, 21) (dual of [390625, 390496, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(597, 390625, F5, 16) (dual of [390625, 390528, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [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(20) ⊂ Ce(15) [i] based on
- discarding factors / shortening the dual code based on linear OA(5137, 390665, F5, 21) (dual of [390665, 390528, 22]-code), using
(137−21, 137, large)-Net in Base 5 — Upper bound on s
There is no (116, 137, large)-net in base 5, because
- 19 times m-reduction [i] would yield (116, 118, large)-net in base 5, but