Best Known (104, 104+17, s)-Nets in Base 16
(104, 104+17, 2098175)-Net over F16 — Constructive and digital
Digital (104, 121, 2098175)-net over F16, using
- (u, u+v)-construction [i] based on
- 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
- net defined by OOA [i] based on linear OOA(1623, 1025, F16, 8, 8) (dual of [(1025, 8), 8177, 9]-NRT-code), using
- digital (81, 98, 2097150)-net over F16, using
- net defined by OOA [i] based on linear OOA(1698, 2097150, F16, 18, 17) (dual of [(2097150, 18), 37748602, 18]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OOA(1698, 8388601, F16, 2, 17) (dual of [(8388601, 2), 16777104, 18]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(1698, 8388602, F16, 2, 17) (dual of [(8388602, 2), 16777106, 18]-NRT-code), using
- trace code [i] based on linear OOA(25649, 4194301, F256, 2, 17) (dual of [(4194301, 2), 8388553, 18]-NRT-code), using
- OOA 2-folding [i] based on linear OA(25649, 8388602, F256, 17) (dual of [8388602, 8388553, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(25649, large, F256, 17) (dual of [large, large−49, 18]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,16], and designed minimum distance d ≥ |I|+1 = 18 [i]
- discarding factors / shortening the dual code based on linear OA(25649, large, F256, 17) (dual of [large, large−49, 18]-code), using
- OOA 2-folding [i] based on linear OA(25649, 8388602, F256, 17) (dual of [8388602, 8388553, 18]-code), using
- trace code [i] based on linear OOA(25649, 4194301, F256, 2, 17) (dual of [(4194301, 2), 8388553, 18]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(1698, 8388602, F16, 2, 17) (dual of [(8388602, 2), 16777106, 18]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OOA(1698, 8388601, F16, 2, 17) (dual of [(8388601, 2), 16777104, 18]-NRT-code), using
- net defined by OOA [i] based on linear OOA(1698, 2097150, F16, 18, 17) (dual of [(2097150, 18), 37748602, 18]-NRT-code), using
- digital (15, 23, 1025)-net over F16, using
(104, 104+17, large)-Net over F16 — Digital
Digital (104, 121, large)-net over F16, using
- 6 times m-reduction [i] based on digital (104, 127, large)-net over F16, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(16127, large, F16, 23) (dual of [large, large−127, 24]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 166−1, defining interval I = [0,22], and designed minimum distance d ≥ |I|+1 = 24 [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(16127, large, F16, 23) (dual of [large, large−127, 24]-code), using
(104, 104+17, large)-Net in Base 16 — Upper bound on s
There is no (104, 121, large)-net in base 16, because
- 15 times m-reduction [i] would yield (104, 106, large)-net in base 16, but