Best Known (113, 113+13, s)-Nets in Base 16
(113, 113+13, 8388602)-Net over F16 — Constructive and digital
Digital (113, 126, 8388602)-net over F16, using
- trace code for nets [i] based on digital (50, 63, 4194301)-net over F256, using
- net defined by OOA [i] based on linear OOA(25663, 4194301, F256, 15, 13) (dual of [(4194301, 15), 62914452, 14]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OOA(25663, large, F256, 3, 13), using
- generalized (u, u+v)-construction [i] based on
- linear OOA(25610, 2796201, F256, 3, 4) (dual of [(2796201, 3), 8388593, 5]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(25610, 4194301, F256, 3, 4) (dual of [(4194301, 3), 12582893, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(25610, 8388602, F256, 4) (dual of [8388602, 8388592, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(25610, large, F256, 4) (dual of [large, large−10, 5]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,3], and designed minimum distance d ≥ |I|+1 = 5 [i]
- discarding factors / shortening the dual code based on linear OA(25610, large, F256, 4) (dual of [large, large−10, 5]-code), using
- OA 2-folding and stacking [i] based on linear OA(25610, 8388602, F256, 4) (dual of [8388602, 8388592, 5]-code), using
- discarding factors / shortening the dual code based on linear OOA(25610, 4194301, F256, 3, 4) (dual of [(4194301, 3), 12582893, 5]-NRT-code), using
- linear OOA(25616, 2796201, F256, 3, 6) (dual of [(2796201, 3), 8388587, 7]-NRT-code), using
- OOA 3-folding [i] based on linear OA(25616, large, F256, 6) (dual of [large, large−16, 7]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,5], and designed minimum distance d ≥ |I|+1 = 7 [i]
- OOA 3-folding [i] based on linear OA(25616, large, F256, 6) (dual of [large, large−16, 7]-code), using
- linear OOA(25637, 2796201, F256, 3, 13) (dual of [(2796201, 3), 8388566, 14]-NRT-code), using
- OOA 3-folding [i] 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]
- OOA 3-folding [i] based on linear OA(25637, large, F256, 13) (dual of [large, large−37, 14]-code), using
- linear OOA(25610, 2796201, F256, 3, 4) (dual of [(2796201, 3), 8388593, 5]-NRT-code), using
- generalized (u, u+v)-construction [i] based on
- OOA 2-folding and stacking with additional row [i] based on linear OOA(25663, large, F256, 3, 13), using
- net defined by OOA [i] based on linear OOA(25663, 4194301, F256, 15, 13) (dual of [(4194301, 15), 62914452, 14]-NRT-code), using
(113, 113+13, large)-Net over F16 — Digital
Digital (113, 126, large)-net over F16, using
- t-expansion [i] based on digital (104, 126, large)-net over F16, using
- 1 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
- 1 times m-reduction [i] based on digital (104, 127, large)-net over F16, using
(113, 113+13, large)-Net in Base 16 — Upper bound on s
There is no (113, 126, large)-net in base 16, because
- 11 times m-reduction [i] would yield (113, 115, large)-net in base 16, but