Best Known (31−16, 31, s)-Nets in Base 81
(31−16, 31, 820)-Net over F81 — Constructive and digital
Digital (15, 31, 820)-net over F81, using
- net defined by OOA [i] based on linear OOA(8131, 820, F81, 16, 16) (dual of [(820, 16), 13089, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(8131, 6560, F81, 16) (dual of [6560, 6529, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(8131, 6561, F81, 16) (dual of [6561, 6530, 17]-code), using
- an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- discarding factors / shortening the dual code based on linear OA(8131, 6561, F81, 16) (dual of [6561, 6530, 17]-code), using
- OA 8-folding and stacking [i] based on linear OA(8131, 6560, F81, 16) (dual of [6560, 6529, 17]-code), using
(31−16, 31, 1872)-Net over F81 — Digital
Digital (15, 31, 1872)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8131, 1872, F81, 3, 16) (dual of [(1872, 3), 5585, 17]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(8131, 2187, F81, 3, 16) (dual of [(2187, 3), 6530, 17]-NRT-code), using
- OOA 3-folding [i] based on linear OA(8131, 6561, F81, 16) (dual of [6561, 6530, 17]-code), using
- an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- OOA 3-folding [i] based on linear OA(8131, 6561, F81, 16) (dual of [6561, 6530, 17]-code), using
- discarding factors / shortening the dual code based on linear OOA(8131, 2187, F81, 3, 16) (dual of [(2187, 3), 6530, 17]-NRT-code), using
(31−16, 31, 1169440)-Net in Base 81 — Upper bound on s
There is no (15, 31, 1169441)-net in base 81, because
- the generalized Rao bound for nets shows that 81m ≥ 145558 172786 418292 315025 635106 931523 685760 786618 777085 005441 > 8131 [i]