Best Known (68−23, 68, s)-Nets in Base 81
(68−23, 68, 48313)-Net over F81 — Constructive and digital
Digital (45, 68, 48313)-net over F81, using
- 811 times duplication [i] based on digital (44, 67, 48313)-net over F81, using
- net defined by OOA [i] based on linear OOA(8167, 48313, F81, 23, 23) (dual of [(48313, 23), 1111132, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(8167, 531444, F81, 23) (dual of [531444, 531377, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(21) [i] based on
- linear OA(8167, 531441, F81, 23) (dual of [531441, 531374, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(8164, 531441, F81, 22) (dual of [531441, 531377, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(810, 3, F81, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(810, s, F81, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(22) ⊂ Ce(21) [i] based on
- OOA 11-folding and stacking with additional row [i] based on linear OA(8167, 531444, F81, 23) (dual of [531444, 531377, 24]-code), using
- net defined by OOA [i] based on linear OOA(8167, 48313, F81, 23, 23) (dual of [(48313, 23), 1111132, 24]-NRT-code), using
(68−23, 68, 206157)-Net over F81 — Digital
Digital (45, 68, 206157)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8168, 206157, F81, 2, 23) (dual of [(206157, 2), 412246, 24]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(8168, 265724, F81, 2, 23) (dual of [(265724, 2), 531380, 24]-NRT-code), using
- OOA 2-folding [i] based on linear OA(8168, 531448, F81, 23) (dual of [531448, 531380, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(8168, 531449, F81, 23) (dual of [531449, 531381, 24]-code), using
- construction X applied to C([0,11]) ⊂ C([0,10]) [i] based on
- linear OA(8167, 531442, F81, 23) (dual of [531442, 531375, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 816−1, defining interval I = [0,11], and minimum distance d ≥ |{−11,−10,…,11}|+1 = 24 (BCH-bound) [i]
- linear OA(8161, 531442, F81, 21) (dual of [531442, 531381, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 816−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(811, 7, F81, 1) (dual of [7, 6, 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
- discarding factors / shortening the dual code based on linear OA(8168, 531449, F81, 23) (dual of [531449, 531381, 24]-code), using
- OOA 2-folding [i] based on linear OA(8168, 531448, F81, 23) (dual of [531448, 531380, 24]-code), using
- discarding factors / shortening the dual code based on linear OOA(8168, 265724, F81, 2, 23) (dual of [(265724, 2), 531380, 24]-NRT-code), using
(68−23, 68, large)-Net in Base 81 — Upper bound on s
There is no (45, 68, large)-net in base 81, because
- 21 times m-reduction [i] would yield (45, 47, large)-net in base 81, but