Best Known (121−13, 121, s)-Nets in Base 16
(121−13, 121, 6116690)-Net over F16 — Constructive and digital
Digital (108, 121, 6116690)-net over F16, using
- generalized (u, u+v)-construction [i] based on
- digital (12, 16, 524290)-net over F16, using
- net defined by OOA [i] based on linear OOA(1616, 524290, F16, 4, 4) (dual of [(524290, 4), 2097144, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(1616, 1048580, F16, 4) (dual of [1048580, 1048564, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(1616, 1048581, F16, 4) (dual of [1048581, 1048565, 5]-code), using
- construction X applied to Ce(3) ⊂ Ce(2) [i] based on
- linear OA(1616, 1048576, F16, 4) (dual of [1048576, 1048560, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(1611, 1048576, F16, 3) (dual of [1048576, 1048565, 4]-code or 1048576-cap in PG(10,16)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(160, 5, F16, 0) (dual of [5, 5, 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(3) ⊂ Ce(2) [i] based on
- discarding factors / shortening the dual code based on linear OA(1616, 1048581, F16, 4) (dual of [1048581, 1048565, 5]-code), using
- OA 2-folding and stacking [i] based on linear OA(1616, 1048580, F16, 4) (dual of [1048580, 1048564, 5]-code), using
- net defined by OOA [i] based on linear OOA(1616, 524290, F16, 4, 4) (dual of [(524290, 4), 2097144, 5]-NRT-code), using
- digital (25, 31, 2796200)-net over F16, using
- s-reduction based on digital (25, 31, 2796201)-net over F16, using
- net defined by OOA [i] based on linear OOA(1631, 2796201, F16, 6, 6) (dual of [(2796201, 6), 16777175, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(1631, large, F16, 6) (dual of [large, large−31, 7]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 166−1, defining interval I = [0,5], and designed minimum distance d ≥ |I|+1 = 7 [i]
- OA 3-folding and stacking [i] based on linear OA(1631, large, F16, 6) (dual of [large, large−31, 7]-code), using
- net defined by OOA [i] based on linear OOA(1631, 2796201, F16, 6, 6) (dual of [(2796201, 6), 16777175, 7]-NRT-code), using
- s-reduction based on digital (25, 31, 2796201)-net over F16, 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, 16, 524290)-net over F16, using
(121−13, 121, large)-Net over F16 — Digital
Digital (108, 121, large)-net over F16, using
- t-expansion [i] based on 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
- 6 times m-reduction [i] based on digital (104, 127, large)-net over F16, using
(121−13, 121, large)-Net in Base 16 — Upper bound on s
There is no (108, 121, large)-net in base 16, because
- 11 times m-reduction [i] would yield (108, 110, large)-net in base 16, but