Best Known (102, 102+15, s)-Nets in Base 4
(102, 102+15, 149801)-Net over F4 — Constructive and digital
Digital (102, 117, 149801)-net over F4, using
- 41 times duplication [i] based on digital (101, 116, 149801)-net over F4, using
- net defined by OOA [i] based on linear OOA(4116, 149801, F4, 15, 15) (dual of [(149801, 15), 2246899, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(4116, 1048608, F4, 15) (dual of [1048608, 1048492, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(4116, 1048611, F4, 15) (dual of [1048611, 1048495, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(10) [i] based on
- linear OA(4111, 1048576, F4, 15) (dual of [1048576, 1048465, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(481, 1048576, F4, 11) (dual of [1048576, 1048495, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(45, 35, F4, 3) (dual of [35, 30, 4]-code or 35-cap in PG(4,4)), using
- discarding factors / shortening the dual code based on linear OA(45, 41, F4, 3) (dual of [41, 36, 4]-code or 41-cap in PG(4,4)), using
- construction X applied to Ce(14) ⊂ Ce(10) [i] based on
- discarding factors / shortening the dual code based on linear OA(4116, 1048611, F4, 15) (dual of [1048611, 1048495, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(4116, 1048608, F4, 15) (dual of [1048608, 1048492, 16]-code), using
- net defined by OOA [i] based on linear OOA(4116, 149801, F4, 15, 15) (dual of [(149801, 15), 2246899, 16]-NRT-code), using
(102, 102+15, 524306)-Net over F4 — Digital
Digital (102, 117, 524306)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4117, 524306, F4, 2, 15) (dual of [(524306, 2), 1048495, 16]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4117, 1048612, F4, 15) (dual of [1048612, 1048495, 16]-code), using
- 1 times code embedding in larger space [i] based on linear OA(4116, 1048611, F4, 15) (dual of [1048611, 1048495, 16]-code), using
- construction X applied to Ce(14) ⊂ Ce(10) [i] based on
- linear OA(4111, 1048576, F4, 15) (dual of [1048576, 1048465, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(481, 1048576, F4, 11) (dual of [1048576, 1048495, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(45, 35, F4, 3) (dual of [35, 30, 4]-code or 35-cap in PG(4,4)), using
- discarding factors / shortening the dual code based on linear OA(45, 41, F4, 3) (dual of [41, 36, 4]-code or 41-cap in PG(4,4)), using
- construction X applied to Ce(14) ⊂ Ce(10) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(4116, 1048611, F4, 15) (dual of [1048611, 1048495, 16]-code), using
- OOA 2-folding [i] based on linear OA(4117, 1048612, F4, 15) (dual of [1048612, 1048495, 16]-code), using
(102, 102+15, large)-Net in Base 4 — Upper bound on s
There is no (102, 117, large)-net in base 4, because
- 13 times m-reduction [i] would yield (102, 104, large)-net in base 4, but