Best Known (118, 118+24, s)-Nets in Base 5
(118, 118+24, 6513)-Net over F5 — Constructive and digital
Digital (118, 142, 6513)-net over F5, using
- 51 times duplication [i] based on digital (117, 141, 6513)-net over F5, using
- net defined by OOA [i] based on linear OOA(5141, 6513, F5, 24, 24) (dual of [(6513, 24), 156171, 25]-NRT-code), using
- OA 12-folding and stacking [i] based on linear OA(5141, 78156, F5, 24) (dual of [78156, 78015, 25]-code), using
- construction XX applied to Ce(23) ⊂ Ce(20) ⊂ Ce(18) [i] based on
- linear OA(5134, 78125, F5, 24) (dual of [78125, 77991, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(5113, 78125, F5, 21) (dual of [78125, 78012, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(5106, 78125, F5, 19) (dual of [78125, 78019, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(53, 27, F5, 2) (dual of [27, 24, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(53, 31, F5, 2) (dual of [31, 28, 3]-code), using
- Hamming code H(3,5) [i]
- discarding factors / shortening the dual code based on linear OA(53, 31, F5, 2) (dual of [31, 28, 3]-code), using
- linear OA(51, 4, F5, 1) (dual of [4, 3, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(51, 5, F5, 1) (dual of [5, 4, 2]-code), using
- Reed–Solomon code RS(4,5) [i]
- discarding factors / shortening the dual code based on linear OA(51, 5, F5, 1) (dual of [5, 4, 2]-code), using
- construction XX applied to Ce(23) ⊂ Ce(20) ⊂ Ce(18) [i] based on
- OA 12-folding and stacking [i] based on linear OA(5141, 78156, F5, 24) (dual of [78156, 78015, 25]-code), using
- net defined by OOA [i] based on linear OOA(5141, 6513, F5, 24, 24) (dual of [(6513, 24), 156171, 25]-NRT-code), using
(118, 118+24, 68306)-Net over F5 — Digital
Digital (118, 142, 68306)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5142, 68306, F5, 24) (dual of [68306, 68164, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(5142, 78161, F5, 24) (dual of [78161, 78019, 25]-code), using
- construction X applied to Ce(23) ⊂ Ce(18) [i] based on
- linear OA(5134, 78125, F5, 24) (dual of [78125, 77991, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(5106, 78125, F5, 19) (dual of [78125, 78019, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(58, 36, F5, 4) (dual of [36, 28, 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(23) ⊂ Ce(18) [i] based on
- discarding factors / shortening the dual code based on linear OA(5142, 78161, F5, 24) (dual of [78161, 78019, 25]-code), using
(118, 118+24, large)-Net in Base 5 — Upper bound on s
There is no (118, 142, large)-net in base 5, because
- 22 times m-reduction [i] would yield (118, 120, large)-net in base 5, but