Best Known (26, 26+25, s)-Nets in Base 81
(26, 26+25, 547)-Net over F81 — Constructive and digital
Digital (26, 51, 547)-net over F81, using
- 811 times duplication [i] based on digital (25, 50, 547)-net over F81, using
- net defined by OOA [i] based on linear OOA(8150, 547, F81, 25, 25) (dual of [(547, 25), 13625, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(8150, 6565, F81, 25) (dual of [6565, 6515, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(8150, 6567, F81, 25) (dual of [6567, 6517, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,11]) [i] based on
- linear OA(8149, 6562, F81, 25) (dual of [6562, 6513, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 814−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- 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(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([0,12]) ⊂ C([0,11]) [i] based on
- discarding factors / shortening the dual code based on linear OA(8150, 6567, F81, 25) (dual of [6567, 6517, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(8150, 6565, F81, 25) (dual of [6565, 6515, 26]-code), using
- net defined by OOA [i] based on linear OOA(8150, 547, F81, 25, 25) (dual of [(547, 25), 13625, 26]-NRT-code), using
(26, 26+25, 2189)-Net over F81 — Digital
Digital (26, 51, 2189)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8151, 2189, F81, 3, 25) (dual of [(2189, 3), 6516, 26]-NRT-code), using
- 811 times duplication [i] based on linear OOA(8150, 2189, F81, 3, 25) (dual of [(2189, 3), 6517, 26]-NRT-code), using
- OOA 3-folding [i] based on linear OA(8150, 6567, F81, 25) (dual of [6567, 6517, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,11]) [i] based on
- linear OA(8149, 6562, F81, 25) (dual of [6562, 6513, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 814−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- 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(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([0,12]) ⊂ C([0,11]) [i] based on
- OOA 3-folding [i] based on linear OA(8150, 6567, F81, 25) (dual of [6567, 6517, 26]-code), using
- 811 times duplication [i] based on linear OOA(8150, 2189, F81, 3, 25) (dual of [(2189, 3), 6517, 26]-NRT-code), using
(26, 26+25, 5919594)-Net in Base 81 — Upper bound on s
There is no (26, 51, 5919595)-net in base 81, because
- 1 times m-reduction [i] would yield (26, 50, 5919595)-net in base 81, but
- the generalized Rao bound for nets shows that 81m ≥ 265614 506445 134468 899970 566945 083561 819027 853177 565744 703395 064188 206457 003526 073412 202287 355201 > 8150 [i]