Best Known (144−19, 144, s)-Nets in Base 5
(144−19, 144, 217018)-Net over F5 — Constructive and digital
Digital (125, 144, 217018)-net over F5, using
- 51 times duplication [i] based on digital (124, 143, 217018)-net over F5, using
- net defined by OOA [i] based on linear OOA(5143, 217018, F5, 19, 19) (dual of [(217018, 19), 4123199, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(5143, 1953163, F5, 19) (dual of [1953163, 1953020, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(5143, 1953168, F5, 19) (dual of [1953168, 1953025, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(13) [i] based on
- linear OA(5136, 1953125, F5, 19) (dual of [1953125, 1952989, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(5100, 1953125, F5, 14) (dual of [1953125, 1953025, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [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(18) ⊂ Ce(13) [i] based on
- discarding factors / shortening the dual code based on linear OA(5143, 1953168, F5, 19) (dual of [1953168, 1953025, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(5143, 1953163, F5, 19) (dual of [1953163, 1953020, 20]-code), using
- net defined by OOA [i] based on linear OOA(5143, 217018, F5, 19, 19) (dual of [(217018, 19), 4123199, 20]-NRT-code), using
(144−19, 144, 1359730)-Net over F5 — Digital
Digital (125, 144, 1359730)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5144, 1359730, F5, 19) (dual of [1359730, 1359586, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(5144, 1953169, F5, 19) (dual of [1953169, 1953025, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(13) [i] based on
- linear OA(5136, 1953125, F5, 19) (dual of [1953125, 1952989, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(5100, 1953125, F5, 14) (dual of [1953125, 1953025, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [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(18) ⊂ Ce(13) [i] based on
- discarding factors / shortening the dual code based on linear OA(5144, 1953169, F5, 19) (dual of [1953169, 1953025, 20]-code), using
(144−19, 144, large)-Net in Base 5 — Upper bound on s
There is no (125, 144, large)-net in base 5, because
- 17 times m-reduction [i] would yield (125, 127, large)-net in base 5, but