Best Known (35, 53, s)-Nets in Base 81
(35, 53, 59049)-Net over F81 — Constructive and digital
Digital (35, 53, 59049)-net over F81, using
- 811 times duplication [i] based on digital (34, 52, 59049)-net over F81, using
- net defined by OOA [i] based on linear OOA(8152, 59049, F81, 18, 18) (dual of [(59049, 18), 1062830, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(8152, 531441, F81, 18) (dual of [531441, 531389, 19]-code), using
- an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- OA 9-folding and stacking [i] based on linear OA(8152, 531441, F81, 18) (dual of [531441, 531389, 19]-code), using
- net defined by OOA [i] based on linear OOA(8152, 59049, F81, 18, 18) (dual of [(59049, 18), 1062830, 19]-NRT-code), using
(35, 53, 247465)-Net over F81 — Digital
Digital (35, 53, 247465)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8153, 247465, F81, 2, 18) (dual of [(247465, 2), 494877, 19]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(8153, 265724, F81, 2, 18) (dual of [(265724, 2), 531395, 19]-NRT-code), using
- OOA 2-folding [i] based on linear OA(8153, 531448, F81, 18) (dual of [531448, 531395, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(15) [i] based on
- linear OA(8152, 531441, F81, 18) (dual of [531441, 531389, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(8146, 531441, F81, 16) (dual of [531441, 531395, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [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 Ce(17) ⊂ Ce(15) [i] based on
- OOA 2-folding [i] based on linear OA(8153, 531448, F81, 18) (dual of [531448, 531395, 19]-code), using
- discarding factors / shortening the dual code based on linear OOA(8153, 265724, F81, 2, 18) (dual of [(265724, 2), 531395, 19]-NRT-code), using
(35, 53, large)-Net in Base 81 — Upper bound on s
There is no (35, 53, large)-net in base 81, because
- 16 times m-reduction [i] would yield (35, 37, large)-net in base 81, but