Best Known (89, 89+31, s)-Nets in Base 16
(89, 89+31, 4370)-Net over F16 — Constructive and digital
Digital (89, 120, 4370)-net over F16, using
- net defined by OOA [i] based on linear OOA(16120, 4370, F16, 31, 31) (dual of [(4370, 31), 135350, 32]-NRT-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(16120, 65551, F16, 31) (dual of [65551, 65431, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(16120, 65554, F16, 31) (dual of [65554, 65434, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(26) [i] based on
- linear OA(16117, 65536, F16, 31) (dual of [65536, 65419, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 65535 = 164−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(16101, 65536, F16, 27) (dual of [65536, 65435, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 65535 = 164−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(163, 18, F16, 3) (dual of [18, 15, 4]-code or 18-arc in PG(2,16) or 18-cap in PG(2,16)), using
- construction X applied to Ce(30) ⊂ Ce(26) [i] based on
- discarding factors / shortening the dual code based on linear OA(16120, 65554, F16, 31) (dual of [65554, 65434, 32]-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(16120, 65551, F16, 31) (dual of [65551, 65431, 32]-code), using
(89, 89+31, 65554)-Net over F16 — Digital
Digital (89, 120, 65554)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(16120, 65554, F16, 31) (dual of [65554, 65434, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(26) [i] based on
- linear OA(16117, 65536, F16, 31) (dual of [65536, 65419, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 65535 = 164−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(16101, 65536, F16, 27) (dual of [65536, 65435, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 65535 = 164−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(163, 18, F16, 3) (dual of [18, 15, 4]-code or 18-arc in PG(2,16) or 18-cap in PG(2,16)), using
- construction X applied to Ce(30) ⊂ Ce(26) [i] based on
(89, 89+31, large)-Net in Base 16 — Upper bound on s
There is no (89, 120, large)-net in base 16, because
- 29 times m-reduction [i] would yield (89, 91, large)-net in base 16, but