Best Known (52, 52+9, s)-Nets in Base 5
(52, 52+9, 97664)-Net over F5 — Constructive and digital
Digital (52, 61, 97664)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (0, 4, 6)-net over F5, using
- net from sequence [i] based on digital (0, 5)-sequence over F5, using
- Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F5 with g(F) = 0 and N(F) ≥ 6, using
- the rational function field F5(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 5)-sequence over F5, using
- digital (48, 57, 97658)-net over F5, using
- net defined by OOA [i] based on linear OOA(557, 97658, F5, 9, 9) (dual of [(97658, 9), 878865, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(557, 390633, F5, 9) (dual of [390633, 390576, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(7) [i] based on
- linear OA(557, 390625, F5, 9) (dual of [390625, 390568, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(549, 390625, F5, 8) (dual of [390625, 390576, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [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(8) ⊂ Ce(7) [i] based on
- OOA 4-folding and stacking with additional row [i] based on linear OA(557, 390633, F5, 9) (dual of [390633, 390576, 10]-code), using
- net defined by OOA [i] based on linear OOA(557, 97658, F5, 9, 9) (dual of [(97658, 9), 878865, 10]-NRT-code), using
- digital (0, 4, 6)-net over F5, using
(52, 52+9, 390654)-Net over F5 — Digital
Digital (52, 61, 390654)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(561, 390654, F5, 9) (dual of [390654, 390593, 10]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(560, 390652, F5, 9) (dual of [390652, 390592, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(5) [i] based on
- linear OA(557, 390625, F5, 9) (dual of [390625, 390568, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(533, 390625, F5, 6) (dual of [390625, 390592, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 390624 = 58−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [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
- construction X applied to Ce(8) ⊂ Ce(5) [i] based on
- linear OA(560, 390653, F5, 8) (dual of [390653, 390593, 9]-code), using Gilbert–Varšamov bound and bm = 560 > Vbs−1(k−1) = 4513 343280 562703 886476 626708 811594 606225 [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(560, 390652, F5, 9) (dual of [390652, 390592, 10]-code), using
- construction X with Varšamov bound [i] based on
(52, 52+9, large)-Net in Base 5 — Upper bound on s
There is no (52, 61, large)-net in base 5, because
- 7 times m-reduction [i] would yield (52, 54, large)-net in base 5, but