Best Known (23−8, 23, s)-Nets in Base 16
(23−8, 23, 1025)-Net over F16 — Constructive and digital
Digital (15, 23, 1025)-net over F16, using
- net defined by OOA [i] based on linear OOA(1623, 1025, F16, 8, 8) (dual of [(1025, 8), 8177, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(1623, 4100, F16, 8) (dual of [4100, 4077, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(1623, 4103, F16, 8) (dual of [4103, 4080, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(5) [i] based on
- linear OA(1622, 4096, F16, 8) (dual of [4096, 4074, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 4095 = 163−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(1616, 4096, F16, 6) (dual of [4096, 4080, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 4095 = 163−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(161, 7, F16, 1) (dual of [7, 6, 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(7) ⊂ Ce(5) [i] based on
- discarding factors / shortening the dual code based on linear OA(1623, 4103, F16, 8) (dual of [4103, 4080, 9]-code), using
- OA 4-folding and stacking [i] based on linear OA(1623, 4100, F16, 8) (dual of [4100, 4077, 9]-code), using
(23−8, 23, 4104)-Net over F16 — Digital
Digital (15, 23, 4104)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1623, 4104, F16, 8) (dual of [4104, 4081, 9]-code), using
- 4 step Varšamov–Edel lengthening with (ri) = (1, 0, 0, 0) [i] based on linear OA(1622, 4099, F16, 8) (dual of [4099, 4077, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(6) [i] based on
- linear OA(1622, 4096, F16, 8) (dual of [4096, 4074, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 4095 = 163−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(1619, 4096, F16, 7) (dual of [4096, 4077, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 4095 = 163−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(160, 3, F16, 0) (dual of [3, 3, 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
- construction X applied to Ce(7) ⊂ Ce(6) [i] based on
- 4 step Varšamov–Edel lengthening with (ri) = (1, 0, 0, 0) [i] based on linear OA(1622, 4099, F16, 8) (dual of [4099, 4077, 9]-code), using
(23−8, 23, 1237801)-Net in Base 16 — Upper bound on s
There is no (15, 23, 1237802)-net in base 16, because
- the generalized Rao bound for nets shows that 16m ≥ 4951 776041 460903 940097 673046 > 1623 [i]