Best Known (234, 234+18, s)-Nets in Base 4
(234, 234+18, 3732365)-Net over F4 — Constructive and digital
Digital (234, 252, 3732365)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (35, 44, 4097)-net over F4, using
- net defined by OOA [i] based on linear OOA(444, 4097, F4, 9, 9) (dual of [(4097, 9), 36829, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(444, 16389, F4, 9) (dual of [16389, 16345, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(444, 16392, F4, 9) (dual of [16392, 16348, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(6) [i] based on
- linear OA(443, 16384, F4, 9) (dual of [16384, 16341, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(436, 16384, F4, 7) (dual of [16384, 16348, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(41, 8, F4, 1) (dual of [8, 7, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(41, s, F4, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(8) ⊂ Ce(6) [i] based on
- discarding factors / shortening the dual code based on linear OA(444, 16392, F4, 9) (dual of [16392, 16348, 10]-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(444, 16389, F4, 9) (dual of [16389, 16345, 10]-code), using
- net defined by OOA [i] based on linear OOA(444, 4097, F4, 9, 9) (dual of [(4097, 9), 36829, 10]-NRT-code), using
- digital (190, 208, 3728268)-net over F4, using
- trace code for nets [i] based on digital (34, 52, 932067)-net over F256, using
- net defined by OOA [i] based on linear OOA(25652, 932067, F256, 18, 18) (dual of [(932067, 18), 16777154, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(25652, large, F256, 18) (dual of [large, large−52, 19]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,17], and designed minimum distance d ≥ |I|+1 = 19 [i]
- OA 9-folding and stacking [i] based on linear OA(25652, large, F256, 18) (dual of [large, large−52, 19]-code), using
- net defined by OOA [i] based on linear OOA(25652, 932067, F256, 18, 18) (dual of [(932067, 18), 16777154, 19]-NRT-code), using
- trace code for nets [i] based on digital (34, 52, 932067)-net over F256, using
- digital (35, 44, 4097)-net over F4, using
(234, 234+18, large)-Net over F4 — Digital
Digital (234, 252, large)-net over F4, using
- t-expansion [i] based on digital (231, 252, large)-net over F4, using
- 5 times m-reduction [i] based on digital (231, 257, large)-net over F4, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(4257, large, F4, 26) (dual of [large, large−257, 27]-code), using
- 28 times code embedding in larger space [i] based on linear OA(4229, large, F4, 26) (dual of [large, large−229, 27]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 412−1, defining interval I = [0,25], and designed minimum distance d ≥ |I|+1 = 27 [i]
- 28 times code embedding in larger space [i] based on linear OA(4229, large, F4, 26) (dual of [large, large−229, 27]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(4257, large, F4, 26) (dual of [large, large−257, 27]-code), using
- 5 times m-reduction [i] based on digital (231, 257, large)-net over F4, using
(234, 234+18, large)-Net in Base 4 — Upper bound on s
There is no (234, 252, large)-net in base 4, because
- 16 times m-reduction [i] would yield (234, 236, large)-net in base 4, but