Best Known (23, 23+23, s)-Nets in Base 81
(23, 23+23, 596)-Net over F81 — Constructive and digital
Digital (23, 46, 596)-net over F81, using
- 811 times duplication [i] based on digital (22, 45, 596)-net over F81, using
- net defined by OOA [i] based on linear OOA(8145, 596, F81, 23, 23) (dual of [(596, 23), 13663, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(8145, 6557, F81, 23) (dual of [6557, 6512, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(8145, 6561, F81, 23) (dual of [6561, 6516, 24]-code), using
- an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- discarding factors / shortening the dual code based on linear OA(8145, 6561, F81, 23) (dual of [6561, 6516, 24]-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(8145, 6557, F81, 23) (dual of [6557, 6512, 24]-code), using
- net defined by OOA [i] based on linear OOA(8145, 596, F81, 23, 23) (dual of [(596, 23), 13663, 24]-NRT-code), using
(23, 23+23, 2058)-Net over F81 — Digital
Digital (23, 46, 2058)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8146, 2058, F81, 3, 23) (dual of [(2058, 3), 6128, 24]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(8146, 2189, F81, 3, 23) (dual of [(2189, 3), 6521, 24]-NRT-code), using
- OOA 3-folding [i] based on linear OA(8146, 6567, F81, 23) (dual of [6567, 6521, 24]-code), using
- construction X applied to C([0,11]) ⊂ C([0,10]) [i] based on
- linear OA(8145, 6562, F81, 23) (dual of [6562, 6517, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 814−1, defining interval I = [0,11], and minimum distance d ≥ |{−11,−10,…,11}|+1 = 24 (BCH-bound) [i]
- linear OA(8141, 6562, F81, 21) (dual of [6562, 6521, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 814−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(811, 5, F81, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(811, s, F81, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,11]) ⊂ C([0,10]) [i] based on
- OOA 3-folding [i] based on linear OA(8146, 6567, F81, 23) (dual of [6567, 6521, 24]-code), using
- discarding factors / shortening the dual code based on linear OOA(8146, 2189, F81, 3, 23) (dual of [(2189, 3), 6521, 24]-NRT-code), using
(23, 23+23, 3938785)-Net in Base 81 — Upper bound on s
There is no (23, 46, 3938786)-net in base 81, because
- 1 times m-reduction [i] would yield (23, 45, 3938786)-net in base 81, but
- the generalized Rao bound for nets shows that 81m ≥ 76 177481 706930 891284 506124 066100 304027 447156 197597 551531 911201 571237 372484 628267 443681 > 8145 [i]