Best Known (88, 88+25, s)-Nets in Base 16
(88, 88+25, 10960)-Net over F16 — Constructive and digital
Digital (88, 113, 10960)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (3, 15, 38)-net over F16, using
- net from sequence [i] based on digital (3, 37)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 3 and N(F) ≥ 38, using
- net from sequence [i] based on digital (3, 37)-sequence over F16, using
- digital (73, 98, 10922)-net over F16, using
- net defined by OOA [i] based on linear OOA(1698, 10922, F16, 25, 25) (dual of [(10922, 25), 272952, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(1698, 131065, F16, 25) (dual of [131065, 130967, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(1698, 131074, F16, 25) (dual of [131074, 130976, 26]-code), using
- trace code [i] based on linear OA(25649, 65537, F256, 25) (dual of [65537, 65488, 26]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- trace code [i] based on linear OA(25649, 65537, F256, 25) (dual of [65537, 65488, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(1698, 131074, F16, 25) (dual of [131074, 130976, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(1698, 131065, F16, 25) (dual of [131065, 130967, 26]-code), using
- net defined by OOA [i] based on linear OOA(1698, 10922, F16, 25, 25) (dual of [(10922, 25), 272952, 26]-NRT-code), using
- digital (3, 15, 38)-net over F16, using
(88, 88+25, 21846)-Net in Base 16 — Constructive
(88, 113, 21846)-net in base 16, using
- 161 times duplication [i] based on (87, 112, 21846)-net in base 16, using
- net defined by OOA [i] based on OOA(16112, 21846, S16, 25, 25), using
- OOA 12-folding and stacking with additional row [i] based on OA(16112, 262153, S16, 25), using
- 1 times code embedding in larger space [i] based on OA(16111, 262152, S16, 25), using
- discarding parts of the base [i] based on linear OA(6474, 262152, F64, 25) (dual of [262152, 262078, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,11]) [i] based on
- linear OA(6473, 262145, F64, 25) (dual of [262145, 262072, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 646−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(6467, 262145, F64, 23) (dual of [262145, 262078, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 646−1, defining interval I = [0,11], and minimum distance d ≥ |{−11,−10,…,11}|+1 = 24 (BCH-bound) [i]
- linear OA(641, 7, F64, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(641, s, F64, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,12]) ⊂ C([0,11]) [i] based on
- discarding parts of the base [i] based on linear OA(6474, 262152, F64, 25) (dual of [262152, 262078, 26]-code), using
- 1 times code embedding in larger space [i] based on OA(16111, 262152, S16, 25), using
- OOA 12-folding and stacking with additional row [i] based on OA(16112, 262153, S16, 25), using
- net defined by OOA [i] based on OOA(16112, 21846, S16, 25, 25), using
(88, 88+25, 305272)-Net over F16 — Digital
Digital (88, 113, 305272)-net over F16, using
(88, 88+25, large)-Net in Base 16 — Upper bound on s
There is no (88, 113, large)-net in base 16, because
- 23 times m-reduction [i] would yield (88, 90, large)-net in base 16, but