Best Known (121−25, 121, s)-Nets in Base 16
(121−25, 121, 87383)-Net over F16 — Constructive and digital
Digital (96, 121, 87383)-net over F16, using
- 161 times duplication [i] based on digital (95, 120, 87383)-net over F16, using
- net defined by OOA [i] based on linear OOA(16120, 87383, F16, 25, 25) (dual of [(87383, 25), 2184455, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(16120, 1048597, F16, 25) (dual of [1048597, 1048477, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(16120, 1048600, F16, 25) (dual of [1048600, 1048480, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(20) [i] based on
- linear OA(16116, 1048576, F16, 25) (dual of [1048576, 1048460, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [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(164, 24, F16, 3) (dual of [24, 20, 4]-code or 24-cap in PG(3,16)), using
- construction X applied to Ce(24) ⊂ Ce(20) [i] based on
- discarding factors / shortening the dual code based on linear OA(16120, 1048600, F16, 25) (dual of [1048600, 1048480, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(16120, 1048597, F16, 25) (dual of [1048597, 1048477, 26]-code), using
- net defined by OOA [i] based on linear OOA(16120, 87383, F16, 25, 25) (dual of [(87383, 25), 2184455, 26]-NRT-code), using
(121−25, 121, 1048602)-Net over F16 — Digital
Digital (96, 121, 1048602)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(16121, 1048602, F16, 25) (dual of [1048602, 1048481, 26]-code), using
- construction XX applied to Ce(24) ⊂ Ce(20) ⊂ Ce(19) [i] based on
- linear OA(16116, 1048576, F16, 25) (dual of [1048576, 1048460, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [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(1691, 1048576, F16, 20) (dual of [1048576, 1048485, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(164, 25, F16, 3) (dual of [25, 21, 4]-code or 25-cap in PG(3,16)), using
- linear OA(160, 1, F16, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(24) ⊂ Ce(20) ⊂ Ce(19) [i] based on
(121−25, 121, large)-Net in Base 16 — Upper bound on s
There is no (96, 121, large)-net in base 16, because
- 23 times m-reduction [i] would yield (96, 98, large)-net in base 16, but