Best Known (42, 64, s)-Nets in Base 81
(42, 64, 48313)-Net over F81 — Constructive and digital
Digital (42, 64, 48313)-net over F81, using
- net defined by OOA [i] based on linear OOA(8164, 48313, F81, 22, 22) (dual of [(48313, 22), 1062822, 23]-NRT-code), using
- OA 11-folding and stacking [i] based on linear OA(8164, 531443, F81, 22) (dual of [531443, 531379, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(8164, 531444, F81, 22) (dual of [531444, 531380, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(20) [i] based on
- 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(8161, 531441, F81, 21) (dual of [531441, 531380, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [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(21) ⊂ Ce(20) [i] based on
- discarding factors / shortening the dual code based on linear OA(8164, 531444, F81, 22) (dual of [531444, 531380, 23]-code), using
- OA 11-folding and stacking [i] based on linear OA(8164, 531443, F81, 22) (dual of [531443, 531379, 23]-code), using
(42, 64, 177148)-Net over F81 — Digital
Digital (42, 64, 177148)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8164, 177148, F81, 3, 22) (dual of [(177148, 3), 531380, 23]-NRT-code), using
- OOA 3-folding [i] based on linear OA(8164, 531444, F81, 22) (dual of [531444, 531380, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(20) [i] based on
- 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(8161, 531441, F81, 21) (dual of [531441, 531380, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [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(21) ⊂ Ce(20) [i] based on
- OOA 3-folding [i] based on linear OA(8164, 531444, F81, 22) (dual of [531444, 531380, 23]-code), using
(42, 64, large)-Net in Base 81 — Upper bound on s
There is no (42, 64, large)-net in base 81, because
- 20 times m-reduction [i] would yield (42, 44, large)-net in base 81, but