Best Known (31−6, 31, s)-Nets in Base 4
(31−6, 31, 5464)-Net over F4 — Constructive and digital
Digital (25, 31, 5464)-net over F4, using
- 41 times duplication [i] based on 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
- net defined by OOA [i] based on linear OOA(430, 5464, F4, 6, 6) (dual of [(5464, 6), 32754, 7]-NRT-code), using
(31−6, 31, 16395)-Net over F4 — Digital
Digital (25, 31, 16395)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(431, 16395, F4, 6) (dual of [16395, 16364, 7]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(429, 16391, F4, 6) (dual of [16391, 16362, 7]-code), using
- construction X 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(40, 7, F4, 0) (dual of [7, 7, 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(5) ⊂ Ce(4) [i] based on
- linear OA(429, 16393, F4, 5) (dual of [16393, 16364, 6]-code), using Gilbert–Varšamov bound and bm = 429 > Vbs−1(k−1) = 243600 353713 024291 [i]
- linear OA(40, 2, F4, 0) (dual of [2, 2, 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 (see above)
- linear OA(429, 16391, F4, 6) (dual of [16391, 16362, 7]-code), using
- construction X with Varšamov bound [i] based on
(31−6, 31, 1008203)-Net in Base 4 — Upper bound on s
There is no (25, 31, 1008204)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 4 611692 555469 741547 > 431 [i]