Best Known (234, 257, s)-Nets in Base 4
(234, 257, 1525200)-Net over F4 — Constructive and digital
Digital (234, 257, 1525200)-net over F4, using
- 43 times duplication [i] based on digital (231, 254, 1525200)-net over F4, using
- trace code for nets [i] based on digital (104, 127, 762600)-net over F16, using
- net defined by OOA [i] based on linear OOA(16127, 762600, F16, 23, 23) (dual of [(762600, 23), 17539673, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(16127, 8388601, F16, 23) (dual of [8388601, 8388474, 24]-code), using
- discarding factors / shortening the dual code 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]
- discarding factors / shortening the dual code based on linear OA(16127, large, F16, 23) (dual of [large, large−127, 24]-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(16127, 8388601, F16, 23) (dual of [8388601, 8388474, 24]-code), using
- net defined by OOA [i] based on linear OOA(16127, 762600, F16, 23, 23) (dual of [(762600, 23), 17539673, 24]-NRT-code), using
- trace code for nets [i] based on digital (104, 127, 762600)-net over F16, using
(234, 257, large)-Net over F4 — Digital
Digital (234, 257, large)-net over F4, using
- t-expansion [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
(234, 257, large)-Net in Base 4 — Upper bound on s
There is no (234, 257, large)-net in base 4, because
- 21 times m-reduction [i] would yield (234, 236, large)-net in base 4, but