Best Known (240−18, 240, s)-Nets in Base 4
(240−18, 240, 3728525)-Net over F4 — Constructive and digital
Digital (222, 240, 3728525)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (23, 32, 257)-net over F4, using
- net defined by OOA [i] based on linear OOA(432, 257, F4, 9, 9) (dual of [(257, 9), 2281, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(432, 1029, F4, 9) (dual of [1029, 997, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(432, 1030, F4, 9) (dual of [1030, 998, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(6) [i] based on
- linear OA(431, 1024, F4, 9) (dual of [1024, 993, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 1023 = 45−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(426, 1024, F4, 7) (dual of [1024, 998, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 1023 = 45−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(41, 6, F4, 1) (dual of [6, 5, 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(432, 1030, F4, 9) (dual of [1030, 998, 10]-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(432, 1029, F4, 9) (dual of [1029, 997, 10]-code), using
- net defined by OOA [i] based on linear OOA(432, 257, F4, 9, 9) (dual of [(257, 9), 2281, 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 (23, 32, 257)-net over F4, using
(240−18, 240, large)-Net over F4 — Digital
Digital (222, 240, large)-net over F4, using
- 7 times m-reduction [i] based on digital (222, 247, large)-net over F4, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(4247, large, F4, 25) (dual of [large, large−247, 26]-code), using
- 30 times code embedding in larger space [i] based on linear OA(4217, large, F4, 25) (dual of [large, large−217, 26]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 424−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- 30 times code embedding in larger space [i] based on linear OA(4217, large, F4, 25) (dual of [large, large−217, 26]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(4247, large, F4, 25) (dual of [large, large−247, 26]-code), using
(240−18, 240, large)-Net in Base 4 — Upper bound on s
There is no (222, 240, large)-net in base 4, because
- 16 times m-reduction [i] would yield (222, 224, large)-net in base 4, but