Best Known (51−12, 51, s)-Nets in Base 16
(51−12, 51, 21848)-Net over F16 — Constructive and digital
Digital (39, 51, 21848)-net over F16, using
- 161 times duplication [i] based on digital (38, 50, 21848)-net over F16, using
- net defined by OOA [i] based on linear OOA(1650, 21848, F16, 12, 12) (dual of [(21848, 12), 262126, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(1650, 131088, F16, 12) (dual of [131088, 131038, 13]-code), using
- trace code [i] based on linear OA(25625, 65544, F256, 12) (dual of [65544, 65519, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(8) [i] based on
- linear OA(25623, 65536, F256, 12) (dual of [65536, 65513, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(25617, 65536, F256, 9) (dual of [65536, 65519, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(2562, 8, F256, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,256)), using
- discarding factors / shortening the dual code based on linear OA(2562, 256, F256, 2) (dual of [256, 254, 3]-code or 256-arc in PG(1,256)), using
- Reed–Solomon code RS(254,256) [i]
- discarding factors / shortening the dual code based on linear OA(2562, 256, F256, 2) (dual of [256, 254, 3]-code or 256-arc in PG(1,256)), using
- construction X applied to Ce(11) ⊂ Ce(8) [i] based on
- trace code [i] based on linear OA(25625, 65544, F256, 12) (dual of [65544, 65519, 13]-code), using
- OA 6-folding and stacking [i] based on linear OA(1650, 131088, F16, 12) (dual of [131088, 131038, 13]-code), using
- net defined by OOA [i] based on linear OOA(1650, 21848, F16, 12, 12) (dual of [(21848, 12), 262126, 13]-NRT-code), using
(51−12, 51, 43691)-Net in Base 16 — Constructive
(39, 51, 43691)-net in base 16, using
- base change [i] based on digital (22, 34, 43691)-net over F64, using
- net defined by OOA [i] based on linear OOA(6434, 43691, F64, 12, 12) (dual of [(43691, 12), 524258, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(6434, 262146, F64, 12) (dual of [262146, 262112, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(6434, 262147, F64, 12) (dual of [262147, 262113, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(10) [i] based on
- linear OA(6434, 262144, F64, 12) (dual of [262144, 262110, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(6431, 262144, F64, 11) (dual of [262144, 262113, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(640, 3, F64, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(640, s, F64, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(11) ⊂ Ce(10) [i] based on
- discarding factors / shortening the dual code based on linear OA(6434, 262147, F64, 12) (dual of [262147, 262113, 13]-code), using
- OA 6-folding and stacking [i] based on linear OA(6434, 262146, F64, 12) (dual of [262146, 262112, 13]-code), using
- net defined by OOA [i] based on linear OOA(6434, 43691, F64, 12, 12) (dual of [(43691, 12), 524258, 13]-NRT-code), using
(51−12, 51, 131090)-Net over F16 — Digital
Digital (39, 51, 131090)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1651, 131090, F16, 12) (dual of [131090, 131039, 13]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(1650, 131088, F16, 12) (dual of [131088, 131038, 13]-code), using
- trace code [i] based on linear OA(25625, 65544, F256, 12) (dual of [65544, 65519, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(8) [i] based on
- linear OA(25623, 65536, F256, 12) (dual of [65536, 65513, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(25617, 65536, F256, 9) (dual of [65536, 65519, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(2562, 8, F256, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,256)), using
- discarding factors / shortening the dual code based on linear OA(2562, 256, F256, 2) (dual of [256, 254, 3]-code or 256-arc in PG(1,256)), using
- Reed–Solomon code RS(254,256) [i]
- discarding factors / shortening the dual code based on linear OA(2562, 256, F256, 2) (dual of [256, 254, 3]-code or 256-arc in PG(1,256)), using
- construction X applied to Ce(11) ⊂ Ce(8) [i] based on
- trace code [i] based on linear OA(25625, 65544, F256, 12) (dual of [65544, 65519, 13]-code), using
- linear OA(1650, 131089, F16, 11) (dual of [131089, 131039, 12]-code), using Gilbert–Varšamov bound and bm = 1650 > Vbs−1(k−1) = 238 030195 374032 030705 784269 317830 412494 336250 444404 605371 [i]
- linear OA(160, 1, F16, 0) (dual of [1, 1, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(160, s, F16, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(1650, 131088, F16, 12) (dual of [131088, 131038, 13]-code), using
- construction X with Varšamov bound [i] based on
(51−12, 51, large)-Net in Base 16 — Upper bound on s
There is no (39, 51, large)-net in base 16, because
- 10 times m-reduction [i] would yield (39, 41, large)-net in base 16, but