Best Known (106−19, 106, s)-Nets in Base 5
(106−19, 106, 8681)-Net over F5 — Constructive and digital
Digital (87, 106, 8681)-net over F5, using
- net defined by OOA [i] based on linear OOA(5106, 8681, F5, 19, 19) (dual of [(8681, 19), 164833, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(5106, 78130, F5, 19) (dual of [78130, 78024, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(5106, 78132, F5, 19) (dual of [78132, 78026, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(17) [i] based on
- 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(599, 78125, F5, 18) (dual of [78125, 78026, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(50, 7, F5, 0) (dual of [7, 7, 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(18) ⊂ Ce(17) [i] based on
- discarding factors / shortening the dual code based on linear OA(5106, 78132, F5, 19) (dual of [78132, 78026, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(5106, 78130, F5, 19) (dual of [78130, 78024, 20]-code), using
(106−19, 106, 39066)-Net over F5 — Digital
Digital (87, 106, 39066)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(5106, 39066, F5, 2, 19) (dual of [(39066, 2), 78026, 20]-NRT-code), using
- OOA 2-folding [i] based on linear OA(5106, 78132, F5, 19) (dual of [78132, 78026, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(17) [i] based on
- 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(599, 78125, F5, 18) (dual of [78125, 78026, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(50, 7, F5, 0) (dual of [7, 7, 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(18) ⊂ Ce(17) [i] based on
- OOA 2-folding [i] based on linear OA(5106, 78132, F5, 19) (dual of [78132, 78026, 20]-code), using
(106−19, 106, large)-Net in Base 5 — Upper bound on s
There is no (87, 106, large)-net in base 5, because
- 17 times m-reduction [i] would yield (87, 89, large)-net in base 5, but