Best Known (203−15, 203, s)-Nets in Base 4
(203−15, 203, 4794851)-Net over F4 — Constructive and digital
Digital (188, 203, 4794851)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (24, 31, 1367)-net over F4, using
- net defined by OOA [i] based on linear OOA(431, 1367, F4, 7, 7) (dual of [(1367, 7), 9538, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(431, 4102, F4, 7) (dual of [4102, 4071, 8]-code), using
- construction X applied to Ce(6) ⊂ Ce(5) [i] based on
- linear OA(431, 4096, F4, 7) (dual of [4096, 4065, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(425, 4096, F4, 6) (dual of [4096, 4071, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 4095 = 46−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(40, 6, F4, 0) (dual of [6, 6, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(40, s, F4, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(6) ⊂ Ce(5) [i] based on
- OOA 3-folding and stacking with additional row [i] based on linear OA(431, 4102, F4, 7) (dual of [4102, 4071, 8]-code), using
- net defined by OOA [i] based on linear OOA(431, 1367, F4, 7, 7) (dual of [(1367, 7), 9538, 8]-NRT-code), using
- digital (157, 172, 4793484)-net over F4, using
- trace code for nets [i] based on digital (28, 43, 1198371)-net over F256, using
- net defined by OOA [i] based on linear OOA(25643, 1198371, F256, 15, 15) (dual of [(1198371, 15), 17975522, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(25643, 8388598, F256, 15) (dual of [8388598, 8388555, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(25643, large, F256, 15) (dual of [large, large−43, 16]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,14], and designed minimum distance d ≥ |I|+1 = 16 [i]
- discarding factors / shortening the dual code based on linear OA(25643, large, F256, 15) (dual of [large, large−43, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(25643, 8388598, F256, 15) (dual of [8388598, 8388555, 16]-code), using
- net defined by OOA [i] based on linear OOA(25643, 1198371, F256, 15, 15) (dual of [(1198371, 15), 17975522, 16]-NRT-code), using
- trace code for nets [i] based on digital (28, 43, 1198371)-net over F256, using
- digital (24, 31, 1367)-net over F4, using
(203−15, 203, large)-Net over F4 — Digital
Digital (188, 203, large)-net over F4, using
- t-expansion [i] based on digital (186, 203, large)-net over F4, using
- 4 times m-reduction [i] based on digital (186, 207, large)-net over F4, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(4207, large, F4, 21) (dual of [large, large−207, 22]-code), using
- 26 times code embedding in larger space [i] based on linear OA(4181, large, F4, 21) (dual of [large, large−181, 22]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 412−1, defining interval I = [0,20], and designed minimum distance d ≥ |I|+1 = 22 [i]
- 26 times code embedding in larger space [i] based on linear OA(4181, large, F4, 21) (dual of [large, large−181, 22]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(4207, large, F4, 21) (dual of [large, large−207, 22]-code), using
- 4 times m-reduction [i] based on digital (186, 207, large)-net over F4, using
(203−15, 203, large)-Net in Base 4 — Upper bound on s
There is no (188, 203, large)-net in base 4, because
- 13 times m-reduction [i] would yield (188, 190, large)-net in base 4, but