Best Known (89, 113, s)-Nets in Base 16
(89, 113, 87382)-Net over F16 — Constructive and digital
Digital (89, 113, 87382)-net over F16, using
- 161 times duplication [i] based on digital (88, 112, 87382)-net over F16, using
- net defined by OOA [i] based on linear OOA(16112, 87382, F16, 24, 24) (dual of [(87382, 24), 2097056, 25]-NRT-code), using
- OA 12-folding and stacking [i] based on linear OA(16112, 1048584, F16, 24) (dual of [1048584, 1048472, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(16112, 1048587, F16, 24) (dual of [1048587, 1048475, 25]-code), using
- construction X applied to Ce(23) ⊂ Ce(21) [i] based on
- linear OA(16111, 1048576, F16, 24) (dual of [1048576, 1048465, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(16101, 1048576, F16, 22) (dual of [1048576, 1048475, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(161, 11, F16, 1) (dual of [11, 10, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(161, s, F16, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(23) ⊂ Ce(21) [i] based on
- discarding factors / shortening the dual code based on linear OA(16112, 1048587, F16, 24) (dual of [1048587, 1048475, 25]-code), using
- OA 12-folding and stacking [i] based on linear OA(16112, 1048584, F16, 24) (dual of [1048584, 1048472, 25]-code), using
- net defined by OOA [i] based on linear OOA(16112, 87382, F16, 24, 24) (dual of [(87382, 24), 2097056, 25]-NRT-code), using
(89, 113, 814369)-Net over F16 — Digital
Digital (89, 113, 814369)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(16113, 814369, F16, 24) (dual of [814369, 814256, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(16113, 1048593, F16, 24) (dual of [1048593, 1048480, 25]-code), using
- construction X applied to Ce(23) ⊂ Ce(20) [i] based on
- linear OA(16111, 1048576, F16, 24) (dual of [1048576, 1048465, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(1696, 1048576, F16, 21) (dual of [1048576, 1048480, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(162, 17, F16, 2) (dual of [17, 15, 3]-code or 17-arc in PG(1,16)), using
- extended Reed–Solomon code RSe(15,16) [i]
- Hamming code H(2,16) [i]
- construction X applied to Ce(23) ⊂ Ce(20) [i] based on
- discarding factors / shortening the dual code based on linear OA(16113, 1048593, F16, 24) (dual of [1048593, 1048480, 25]-code), using
(89, 113, large)-Net in Base 16 — Upper bound on s
There is no (89, 113, large)-net in base 16, because
- 22 times m-reduction [i] would yield (89, 91, large)-net in base 16, but