Best Known (18, 18+6, s)-Nets in Base 16
(18, 18+6, 43694)-Net over F16 — Constructive and digital
Digital (18, 24, 43694)-net over F16, using
- net defined by OOA [i] based on linear OOA(1624, 43694, F16, 6, 6) (dual of [(43694, 6), 262140, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(1624, 131082, F16, 6) (dual of [131082, 131058, 7]-code), using
- trace code [i] based on linear OA(25612, 65541, F256, 6) (dual of [65541, 65529, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(3) [i] based on
- linear OA(25611, 65536, F256, 6) (dual of [65536, 65525, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(2567, 65536, F256, 4) (dual of [65536, 65529, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(2561, 5, F256, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(2561, s, F256, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(5) ⊂ Ce(3) [i] based on
- trace code [i] based on linear OA(25612, 65541, F256, 6) (dual of [65541, 65529, 7]-code), using
- OA 3-folding and stacking [i] based on linear OA(1624, 131082, F16, 6) (dual of [131082, 131058, 7]-code), using
(18, 18+6, 87382)-Net in Base 16 — Constructive
(18, 24, 87382)-net in base 16, using
- base change [i] based on digital (10, 16, 87382)-net over F64, using
- net defined by OOA [i] based on linear OOA(6416, 87382, F64, 6, 6) (dual of [(87382, 6), 524276, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(6416, 262146, F64, 6) (dual of [262146, 262130, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(6416, 262147, F64, 6) (dual of [262147, 262131, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(4) [i] based on
- linear OA(6416, 262144, F64, 6) (dual of [262144, 262128, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(6413, 262144, F64, 5) (dual of [262144, 262131, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [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(5) ⊂ Ce(4) [i] based on
- discarding factors / shortening the dual code based on linear OA(6416, 262147, F64, 6) (dual of [262147, 262131, 7]-code), using
- OA 3-folding and stacking [i] based on linear OA(6416, 262146, F64, 6) (dual of [262146, 262130, 7]-code), using
- net defined by OOA [i] based on linear OOA(6416, 87382, F64, 6, 6) (dual of [(87382, 6), 524276, 7]-NRT-code), using
(18, 18+6, 131082)-Net over F16 — Digital
Digital (18, 24, 131082)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1624, 131082, F16, 6) (dual of [131082, 131058, 7]-code), using
- trace code [i] based on linear OA(25612, 65541, F256, 6) (dual of [65541, 65529, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(3) [i] based on
- linear OA(25611, 65536, F256, 6) (dual of [65536, 65525, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(2567, 65536, F256, 4) (dual of [65536, 65529, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(2561, 5, F256, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(2561, s, F256, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(5) ⊂ Ce(3) [i] based on
- trace code [i] based on linear OA(25612, 65541, F256, 6) (dual of [65541, 65529, 7]-code), using
(18, 18+6, 208394)-Net in Base 16
(18, 24, 208394)-net in base 16, using
- base change [i] based on digital (10, 16, 208394)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6416, 208394, F64, 6) (dual of [208394, 208378, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(6416, 262144, F64, 6) (dual of [262144, 262128, 7]-code), using
- an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- discarding factors / shortening the dual code based on linear OA(6416, 262144, F64, 6) (dual of [262144, 262128, 7]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6416, 208394, F64, 6) (dual of [208394, 208378, 7]-code), using
(18, 18+6, large)-Net in Base 16 — Upper bound on s
There is no (18, 24, large)-net in base 16, because
- 4 times m-reduction [i] would yield (18, 20, large)-net in base 16, but