Best Known (122−13, 122, s)-Nets in Base 16
(122−13, 122, 6116694)-Net over F16 — Constructive and digital
Digital (109, 122, 6116694)-net over F16, using
- generalized (u, u+v)-construction [i] based on
- digital (13, 17, 524294)-net over F16, using
- net defined by OOA [i] based on linear OOA(1617, 524294, F16, 4, 4) (dual of [(524294, 4), 2097159, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(1617, 1048588, F16, 4) (dual of [1048588, 1048571, 5]-code), using
- construction X4 applied to Ce(3) ⊂ Ce(1) [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(166, 1048576, F16, 2) (dual of [1048576, 1048570, 3]-code), using an extension Ce(1) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,1], and designed minimum distance d ≥ |I|+1 = 2 [i]
- linear OA(1611, 12, F16, 11) (dual of [12, 1, 12]-code or 12-arc in PG(10,16)), using
- dual of repetition code with length 12 [i]
- linear OA(161, 12, F16, 1) (dual of [12, 11, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(161, 16, F16, 1) (dual of [16, 15, 2]-code), using
- Reed–Solomon code RS(15,16) [i]
- discarding factors / shortening the dual code based on linear OA(161, 16, F16, 1) (dual of [16, 15, 2]-code), using
- construction X4 applied to Ce(3) ⊂ Ce(1) [i] based on
- OA 2-folding and stacking [i] based on linear OA(1617, 1048588, F16, 4) (dual of [1048588, 1048571, 5]-code), using
- net defined by OOA [i] based on linear OOA(1617, 524294, F16, 4, 4) (dual of [(524294, 4), 2097159, 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 (13, 17, 524294)-net over F16, using
(122−13, 122, large)-Net over F16 — Digital
Digital (109, 122, large)-net over F16, using
- t-expansion [i] based on digital (104, 122, large)-net over F16, using
- 5 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
- 5 times m-reduction [i] based on digital (104, 127, large)-net over F16, using
(122−13, 122, large)-Net in Base 16 — Upper bound on s
There is no (109, 122, large)-net in base 16, because
- 11 times m-reduction [i] would yield (109, 111, large)-net in base 16, but