Best Known (126−18, 126, s)-Nets in Base 5
(126−18, 126, 43421)-Net over F5 — Constructive and digital
Digital (108, 126, 43421)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (4, 13, 18)-net over F5, using
- net from sequence [i] based on digital (4, 17)-sequence over F5, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F5 with g(F) = 4 and N(F) ≥ 18, using
- net from sequence [i] based on digital (4, 17)-sequence over F5, using
- digital (95, 113, 43403)-net over F5, using
- net defined by OOA [i] based on linear OOA(5113, 43403, F5, 18, 18) (dual of [(43403, 18), 781141, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(5113, 390627, F5, 18) (dual of [390627, 390514, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(5113, 390633, F5, 18) (dual of [390633, 390520, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(16) [i] based on
- linear OA(5113, 390625, F5, 18) (dual of [390625, 390512, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [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(50, 8, F5, 0) (dual of [8, 8, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(50, s, F5, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(17) ⊂ Ce(16) [i] based on
- discarding factors / shortening the dual code based on linear OA(5113, 390633, F5, 18) (dual of [390633, 390520, 19]-code), using
- OA 9-folding and stacking [i] based on linear OA(5113, 390627, F5, 18) (dual of [390627, 390514, 19]-code), using
- net defined by OOA [i] based on linear OOA(5113, 43403, F5, 18, 18) (dual of [(43403, 18), 781141, 19]-NRT-code), using
- digital (4, 13, 18)-net over F5, using
(126−18, 126, 390687)-Net over F5 — Digital
Digital (108, 126, 390687)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5126, 390687, F5, 18) (dual of [390687, 390561, 19]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(5125, 390685, F5, 18) (dual of [390685, 390560, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(10) [i] based on
- linear OA(5113, 390625, F5, 18) (dual of [390625, 390512, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(565, 390625, F5, 11) (dual of [390625, 390560, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(512, 60, F5, 6) (dual of [60, 48, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(512, 62, F5, 6) (dual of [62, 50, 7]-code), using
- the cyclic code C(A) with length 62 | 53−1, defining set A = {4,8,11,17}, and minimum distance d ≥ |{8,11,14,…,23}|+1 = 7 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(512, 62, F5, 6) (dual of [62, 50, 7]-code), using
- construction X applied to Ce(17) ⊂ Ce(10) [i] based on
- linear OA(5125, 390686, F5, 17) (dual of [390686, 390561, 18]-code), using Gilbert–Varšamov bound and bm = 5125 > Vbs−1(k−1) = 60 455958 735334 502993 880184 891632 011854 498948 569514 196087 722235 075298 406401 318037 030485 [i]
- linear OA(50, 1, F5, 0) (dual of [1, 1, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(50, s, F5, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(5125, 390685, F5, 18) (dual of [390685, 390560, 19]-code), using
- construction X with Varšamov bound [i] based on
(126−18, 126, large)-Net in Base 5 — Upper bound on s
There is no (108, 126, large)-net in base 5, because
- 16 times m-reduction [i] would yield (108, 110, large)-net in base 5, but