Best Known (81−21, 81, s)-Nets in Base 5
(81−21, 81, 312)-Net over F5 — Constructive and digital
Digital (60, 81, 312)-net over F5, using
- net defined by OOA [i] based on linear OOA(581, 312, F5, 21, 21) (dual of [(312, 21), 6471, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(581, 3121, F5, 21) (dual of [3121, 3040, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(581, 3125, F5, 21) (dual of [3125, 3044, 22]-code), using
- an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- discarding factors / shortening the dual code based on linear OA(581, 3125, F5, 21) (dual of [3125, 3044, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(581, 3121, F5, 21) (dual of [3121, 3040, 22]-code), using
(81−21, 81, 1726)-Net over F5 — Digital
Digital (60, 81, 1726)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(581, 1726, F5, 21) (dual of [1726, 1645, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(581, 3125, F5, 21) (dual of [3125, 3044, 22]-code), using
- an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- discarding factors / shortening the dual code based on linear OA(581, 3125, F5, 21) (dual of [3125, 3044, 22]-code), using
(81−21, 81, 442251)-Net in Base 5 — Upper bound on s
There is no (60, 81, 442252)-net in base 5, because
- 1 times m-reduction [i] would yield (60, 80, 442252)-net in base 5, but
- the generalized Rao bound for nets shows that 5m ≥ 82 718695 326423 383210 270531 795000 689053 028514 747746 441025 > 580 [i]