Best Known (30−6, 30, s)-Nets in Base 4
(30−6, 30, 5464)-Net over F4 — Constructive and digital
Digital (24, 30, 5464)-net over F4, using
- net defined by OOA [i] based on linear OOA(430, 5464, F4, 6, 6) (dual of [(5464, 6), 32754, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(430, 16392, F4, 6) (dual of [16392, 16362, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(430, 16393, F4, 6) (dual of [16393, 16363, 7]-code), using
- construction X4 applied to Ce(5) ⊂ Ce(4) [i] based on
- linear OA(429, 16384, F4, 6) (dual of [16384, 16355, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(422, 16384, F4, 5) (dual of [16384, 16362, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(48, 9, F4, 8) (dual of [9, 1, 9]-code or 9-arc in PG(7,4)), using
- dual of repetition code with length 9 [i]
- linear OA(41, 9, F4, 1) (dual of [9, 8, 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 X4 applied to Ce(5) ⊂ Ce(4) [i] based on
- discarding factors / shortening the dual code based on linear OA(430, 16393, F4, 6) (dual of [16393, 16363, 7]-code), using
- OA 3-folding and stacking [i] based on linear OA(430, 16392, F4, 6) (dual of [16392, 16362, 7]-code), using
(30−6, 30, 16393)-Net over F4 — Digital
Digital (24, 30, 16393)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(430, 16393, F4, 6) (dual of [16393, 16363, 7]-code), using
- construction X4 applied to Ce(5) ⊂ Ce(4) [i] based on
- linear OA(429, 16384, F4, 6) (dual of [16384, 16355, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(422, 16384, F4, 5) (dual of [16384, 16362, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(48, 9, F4, 8) (dual of [9, 1, 9]-code or 9-arc in PG(7,4)), using
- dual of repetition code with length 9 [i]
- linear OA(41, 9, F4, 1) (dual of [9, 8, 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 X4 applied to Ce(5) ⊂ Ce(4) [i] based on
(30−6, 30, 635127)-Net in Base 4 — Upper bound on s
There is no (24, 30, 635128)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 1 152923 242245 846085 > 430 [i]