Best Known (79, 79+13, s)-Nets in Base 16
(79, 79+13, 2797569)-Net over F16 — Constructive and digital
Digital (79, 92, 2797569)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (12, 18, 1369)-net over F16, using
- net defined by OOA [i] based on linear OOA(1618, 1369, F16, 6, 6) (dual of [(1369, 6), 8196, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(1618, 4107, F16, 6) (dual of [4107, 4089, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(2) [i] based on
- 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(167, 4096, F16, 3) (dual of [4096, 4089, 4]-code or 4096-cap in PG(6,16)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 4095 = 163−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(162, 11, F16, 2) (dual of [11, 9, 3]-code or 11-arc in PG(1,16)), using
- discarding factors / shortening the dual code based on linear OA(162, 16, F16, 2) (dual of [16, 14, 3]-code or 16-arc in PG(1,16)), using
- Reed–Solomon code RS(14,16) [i]
- discarding factors / shortening the dual code based on linear OA(162, 16, F16, 2) (dual of [16, 14, 3]-code or 16-arc in PG(1,16)), using
- construction X applied to Ce(5) ⊂ Ce(2) [i] based on
- OA 3-folding and stacking [i] based on linear OA(1618, 4107, F16, 6) (dual of [4107, 4089, 7]-code), using
- net defined by OOA [i] based on linear OOA(1618, 1369, F16, 6, 6) (dual of [(1369, 6), 8196, 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 (12, 18, 1369)-net over F16, using
(79, 79+13, large)-Net over F16 — Digital
Digital (79, 92, large)-net over F16, using
- t-expansion [i] based on digital (75, 92, large)-net over F16, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(1692, large, F16, 17) (dual of [large, large−92, 18]-code), using
- 1 times code embedding in larger space [i] based on linear OA(1691, large, F16, 17) (dual of [large, large−91, 18]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 166−1, defining interval I = [0,16], and designed minimum distance d ≥ |I|+1 = 18 [i]
- 1 times code embedding in larger space [i] based on linear OA(1691, large, F16, 17) (dual of [large, large−91, 18]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(1692, large, F16, 17) (dual of [large, large−92, 18]-code), using
(79, 79+13, large)-Net in Base 16 — Upper bound on s
There is no (79, 92, large)-net in base 16, because
- 11 times m-reduction [i] would yield (79, 81, large)-net in base 16, but