Best Known (84, 84+13, s)-Nets in Base 16
(84, 84+13, 2839892)-Net over F16 — Constructive and digital
Digital (84, 97, 2839892)-net over F16, using
- 161 times duplication [i] based on digital (83, 96, 2839892)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (16, 22, 43692)-net over F16, using
- net defined by OOA [i] based on linear OOA(1622, 43692, F16, 6, 6) (dual of [(43692, 6), 262130, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(1622, 131076, F16, 6) (dual of [131076, 131054, 7]-code), using
- trace code [i] based on linear OA(25611, 65538, F256, 6) (dual of [65538, 65527, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(4) [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(2569, 65536, F256, 5) (dual of [65536, 65527, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(2560, 2, F256, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(2560, s, F256, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(5) ⊂ Ce(4) [i] based on
- trace code [i] based on linear OA(25611, 65538, F256, 6) (dual of [65538, 65527, 7]-code), using
- OA 3-folding and stacking [i] based on linear OA(1622, 131076, F16, 6) (dual of [131076, 131054, 7]-code), using
- net defined by OOA [i] based on linear OOA(1622, 43692, F16, 6, 6) (dual of [(43692, 6), 262130, 7]-NRT-code), using
- digital (61, 74, 2796200)-net over F16, using
- net defined by OOA [i] based on linear OOA(1674, 2796200, F16, 14, 13) (dual of [(2796200, 14), 39146726, 14]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OOA(1674, 8388601, F16, 2, 13) (dual of [(8388601, 2), 16777128, 14]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(1674, 8388602, F16, 2, 13) (dual of [(8388602, 2), 16777130, 14]-NRT-code), using
- trace code [i] based on linear OOA(25637, 4194301, F256, 2, 13) (dual of [(4194301, 2), 8388565, 14]-NRT-code), using
- OOA 2-folding [i] based on linear OA(25637, 8388602, F256, 13) (dual of [8388602, 8388565, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(25637, large, F256, 13) (dual of [large, large−37, 14]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,12], and designed minimum distance d ≥ |I|+1 = 14 [i]
- discarding factors / shortening the dual code based on linear OA(25637, large, F256, 13) (dual of [large, large−37, 14]-code), using
- OOA 2-folding [i] based on linear OA(25637, 8388602, F256, 13) (dual of [8388602, 8388565, 14]-code), using
- trace code [i] based on linear OOA(25637, 4194301, F256, 2, 13) (dual of [(4194301, 2), 8388565, 14]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(1674, 8388602, F16, 2, 13) (dual of [(8388602, 2), 16777130, 14]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OOA(1674, 8388601, F16, 2, 13) (dual of [(8388601, 2), 16777128, 14]-NRT-code), using
- net defined by OOA [i] based on linear OOA(1674, 2796200, F16, 14, 13) (dual of [(2796200, 14), 39146726, 14]-NRT-code), using
- digital (16, 22, 43692)-net over F16, using
- (u, u+v)-construction [i] based on
(84, 84+13, large)-Net over F16 — Digital
Digital (84, 97, large)-net over F16, using
- t-expansion [i] based on digital (80, 97, large)-net over F16, using
- 1 times m-reduction [i] based on digital (80, 98, large)-net over F16, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(1698, large, F16, 18) (dual of [large, large−98, 19]-code), using
- 1 times code embedding in larger space [i] based on linear OA(1697, large, F16, 18) (dual of [large, large−97, 19]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 166−1, defining interval I = [0,17], and designed minimum distance d ≥ |I|+1 = 19 [i]
- 1 times code embedding in larger space [i] based on linear OA(1697, large, F16, 18) (dual of [large, large−97, 19]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(1698, large, F16, 18) (dual of [large, large−98, 19]-code), using
- 1 times m-reduction [i] based on digital (80, 98, large)-net over F16, using
(84, 84+13, large)-Net in Base 16 — Upper bound on s
There is no (84, 97, large)-net in base 16, because
- 11 times m-reduction [i] would yield (84, 86, large)-net in base 16, but