Best Known (146−22, 146, s)-Nets in Base 5
(146−22, 146, 35515)-Net over F5 — Constructive and digital
Digital (124, 146, 35515)-net over F5, using
- 51 times duplication [i] based on digital (123, 145, 35515)-net over F5, using
- net defined by OOA [i] based on linear OOA(5145, 35515, F5, 22, 22) (dual of [(35515, 22), 781185, 23]-NRT-code), using
- OA 11-folding and stacking [i] based on linear OA(5145, 390665, F5, 22) (dual of [390665, 390520, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(16) [i] based on
- linear OA(5137, 390625, F5, 22) (dual of [390625, 390488, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(5105, 390625, F5, 17) (dual of [390625, 390520, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [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(21) ⊂ Ce(16) [i] based on
- OA 11-folding and stacking [i] based on linear OA(5145, 390665, F5, 22) (dual of [390665, 390520, 23]-code), using
- net defined by OOA [i] based on linear OOA(5145, 35515, F5, 22, 22) (dual of [(35515, 22), 781185, 23]-NRT-code), using
(146−22, 146, 242523)-Net over F5 — Digital
Digital (124, 146, 242523)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5146, 242523, F5, 22) (dual of [242523, 242377, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(5146, 390669, F5, 22) (dual of [390669, 390523, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(15) [i] based on
- linear OA(5137, 390625, F5, 22) (dual of [390625, 390488, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [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(59, 44, F5, 5) (dual of [44, 35, 6]-code), using
- construction X applied to Ce(21) ⊂ Ce(15) [i] based on
- discarding factors / shortening the dual code based on linear OA(5146, 390669, F5, 22) (dual of [390669, 390523, 23]-code), using
(146−22, 146, large)-Net in Base 5 — Upper bound on s
There is no (124, 146, large)-net in base 5, because
- 20 times m-reduction [i] would yield (124, 126, large)-net in base 5, but