Best Known (91, 106, s)-Nets in Base 16
(91, 106, 2398110)-Net over F16 — Constructive and digital
Digital (91, 106, 2398110)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (13, 20, 1368)-net over F16, using
- net defined by OOA [i] based on linear OOA(1620, 1368, F16, 7, 7) (dual of [(1368, 7), 9556, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(1620, 4105, F16, 7) (dual of [4105, 4085, 8]-code), using
- construction X4 applied to C([0,3]) ⊂ C([0,2]) [i] based on
- linear OA(1619, 4097, F16, 7) (dual of [4097, 4078, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 166−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- linear OA(1613, 4097, F16, 5) (dual of [4097, 4084, 6]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 166−1, defining interval I = [0,2], and minimum distance d ≥ |{−2,−1,0,1,2}|+1 = 6 (BCH-bound) [i]
- linear OA(167, 8, F16, 7) (dual of [8, 1, 8]-code or 8-arc in PG(6,16)), using
- dual of repetition code with length 8 [i]
- linear OA(161, 8, F16, 1) (dual of [8, 7, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(161, 16, F16, 1) (dual of [16, 15, 2]-code), using
- Reed–Solomon code RS(15,16) [i]
- discarding factors / shortening the dual code based on linear OA(161, 16, F16, 1) (dual of [16, 15, 2]-code), using
- construction X4 applied to C([0,3]) ⊂ C([0,2]) [i] based on
- OOA 3-folding and stacking with additional row [i] based on linear OA(1620, 4105, F16, 7) (dual of [4105, 4085, 8]-code), using
- net defined by OOA [i] based on linear OOA(1620, 1368, F16, 7, 7) (dual of [(1368, 7), 9556, 8]-NRT-code), using
- digital (71, 86, 2396742)-net over F16, using
- trace code for nets [i] based on digital (28, 43, 1198371)-net over F256, using
- net defined by OOA [i] based on linear OOA(25643, 1198371, F256, 15, 15) (dual of [(1198371, 15), 17975522, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(25643, 8388598, F256, 15) (dual of [8388598, 8388555, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(25643, large, F256, 15) (dual of [large, large−43, 16]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,14], and designed minimum distance d ≥ |I|+1 = 16 [i]
- discarding factors / shortening the dual code based on linear OA(25643, large, F256, 15) (dual of [large, large−43, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(25643, 8388598, F256, 15) (dual of [8388598, 8388555, 16]-code), using
- net defined by OOA [i] based on linear OOA(25643, 1198371, F256, 15, 15) (dual of [(1198371, 15), 17975522, 16]-NRT-code), using
- trace code for nets [i] based on digital (28, 43, 1198371)-net over F256, using
- digital (13, 20, 1368)-net over F16, using
(91, 106, large)-Net over F16 — Digital
Digital (91, 106, large)-net over F16, using
- t-expansion [i] based on digital (89, 106, large)-net over F16, using
- 3 times m-reduction [i] based on digital (89, 109, large)-net over F16, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(16109, large, F16, 20) (dual of [large, large−109, 21]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 166−1, defining interval I = [0,19], and designed minimum distance d ≥ |I|+1 = 21 [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(16109, large, F16, 20) (dual of [large, large−109, 21]-code), using
- 3 times m-reduction [i] based on digital (89, 109, large)-net over F16, using
(91, 106, large)-Net in Base 16 — Upper bound on s
There is no (91, 106, large)-net in base 16, because
- 13 times m-reduction [i] would yield (91, 93, large)-net in base 16, but