Best Known (21, 21+6, s)-Nets in Base 16
(21, 21+6, 349529)-Net over F16 — Constructive and digital
Digital (21, 27, 349529)-net over F16, using
- net defined by OOA [i] based on linear OOA(1627, 349529, F16, 6, 6) (dual of [(349529, 6), 2097147, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(1627, 1048587, F16, 6) (dual of [1048587, 1048560, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(3) [i] based on
- linear OA(1626, 1048576, F16, 6) (dual of [1048576, 1048550, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(1616, 1048576, F16, 4) (dual of [1048576, 1048560, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(161, 11, F16, 1) (dual of [11, 10, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(161, s, F16, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(5) ⊂ Ce(3) [i] based on
- OA 3-folding and stacking [i] based on linear OA(1627, 1048587, F16, 6) (dual of [1048587, 1048560, 7]-code), using
(21, 21+6, 1048588)-Net over F16 — Digital
Digital (21, 27, 1048588)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1627, 1048588, F16, 6) (dual of [1048588, 1048561, 7]-code), using
- construction X4 applied to Ce(5) ⊂ Ce(3) [i] based on
- linear OA(1626, 1048576, F16, 6) (dual of [1048576, 1048550, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(1616, 1048576, F16, 4) (dual of [1048576, 1048560, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(1611, 12, F16, 11) (dual of [12, 1, 12]-code or 12-arc in PG(10,16)), using
- dual of repetition code with length 12 [i]
- linear OA(161, 12, F16, 1) (dual of [12, 11, 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 Ce(5) ⊂ Ce(3) [i] based on
(21, 21+6, large)-Net in Base 16 — Upper bound on s
There is no (21, 27, large)-net in base 16, because
- 4 times m-reduction [i] would yield (21, 23, large)-net in base 16, but