Best Known (257−43, 257, s)-Nets in Base 4
(257−43, 257, 3121)-Net over F4 — Constructive and digital
Digital (214, 257, 3121)-net over F4, using
- net defined by OOA [i] based on linear OOA(4257, 3121, F4, 43, 43) (dual of [(3121, 43), 133946, 44]-NRT-code), using
- OOA 21-folding and stacking with additional row [i] based on linear OA(4257, 65542, F4, 43) (dual of [65542, 65285, 44]-code), using
- discarding factors / shortening the dual code based on linear OA(4257, 65544, F4, 43) (dual of [65544, 65287, 44]-code), using
- construction X applied to Ce(42) ⊂ Ce(41) [i] based on
- linear OA(4257, 65536, F4, 43) (dual of [65536, 65279, 44]-code), using an extension Ce(42) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,42], and designed minimum distance d ≥ |I|+1 = 43 [i]
- linear OA(4249, 65536, F4, 42) (dual of [65536, 65287, 43]-code), using an extension Ce(41) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,41], and designed minimum distance d ≥ |I|+1 = 42 [i]
- linear OA(40, 8, F4, 0) (dual of [8, 8, 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(42) ⊂ Ce(41) [i] based on
- discarding factors / shortening the dual code based on linear OA(4257, 65544, F4, 43) (dual of [65544, 65287, 44]-code), using
- OOA 21-folding and stacking with additional row [i] based on linear OA(4257, 65542, F4, 43) (dual of [65542, 65285, 44]-code), using
(257−43, 257, 32772)-Net over F4 — Digital
Digital (214, 257, 32772)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4257, 32772, F4, 2, 43) (dual of [(32772, 2), 65287, 44]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4257, 65544, F4, 43) (dual of [65544, 65287, 44]-code), using
- construction X applied to Ce(42) ⊂ Ce(41) [i] based on
- linear OA(4257, 65536, F4, 43) (dual of [65536, 65279, 44]-code), using an extension Ce(42) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,42], and designed minimum distance d ≥ |I|+1 = 43 [i]
- linear OA(4249, 65536, F4, 42) (dual of [65536, 65287, 43]-code), using an extension Ce(41) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,41], and designed minimum distance d ≥ |I|+1 = 42 [i]
- linear OA(40, 8, F4, 0) (dual of [8, 8, 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(42) ⊂ Ce(41) [i] based on
- OOA 2-folding [i] based on linear OA(4257, 65544, F4, 43) (dual of [65544, 65287, 44]-code), using
(257−43, 257, large)-Net in Base 4 — Upper bound on s
There is no (214, 257, large)-net in base 4, because
- 41 times m-reduction [i] would yield (214, 216, large)-net in base 4, but