Best Known (53, 53+9, s)-Nets in Base 5
(53, 53+9, 97670)-Net over F5 — Constructive and digital
Digital (53, 62, 97670)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (1, 5, 12)-net 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
(53, 53+9, 390656)-Net over F5 — Digital
Digital (53, 62, 390656)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(562, 390656, F5, 9) (dual of [390656, 390594, 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, 390654, F5, 8) (dual of [390654, 390594, 9]-code), using Gilbert–Varšamov bound and bm = 560 > Vbs−1(k−1) = 4513 424155 274706 243467 014223 905055 097045 [i]
- linear OA(50, 2, F5, 0) (dual of [2, 2, 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
(53, 53+9, large)-Net in Base 5 — Upper bound on s
There is no (53, 62, large)-net in base 5, because
- 7 times m-reduction [i] would yield (53, 55, large)-net in base 5, but