Best Known (28−7, 28, s)-Nets in Base 5
(28−7, 28, 1044)-Net over F5 — Constructive and digital
Digital (21, 28, 1044)-net over F5, using
- net defined by OOA [i] based on linear OOA(528, 1044, F5, 7, 7) (dual of [(1044, 7), 7280, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(528, 3133, F5, 7) (dual of [3133, 3105, 8]-code), using
- construction XX applied to Ce(6) ⊂ Ce(5) ⊂ Ce(3) [i] based on
- linear OA(526, 3125, F5, 7) (dual of [3125, 3099, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(521, 3125, F5, 6) (dual of [3125, 3104, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(516, 3125, F5, 4) (dual of [3125, 3109, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(50, 6, F5, 0) (dual of [6, 6, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(50, s, F5, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(51, 2, F5, 1) (dual of [2, 1, 2]-code), using
- dual of repetition code with length 2 [i]
- construction XX applied to Ce(6) ⊂ Ce(5) ⊂ Ce(3) [i] based on
- OOA 3-folding and stacking with additional row [i] based on linear OA(528, 3133, F5, 7) (dual of [3133, 3105, 8]-code), using
(28−7, 28, 3136)-Net over F5 — Digital
Digital (21, 28, 3136)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(528, 3136, F5, 7) (dual of [3136, 3108, 8]-code), using
- 4 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 0) [i] based on linear OA(526, 3130, F5, 7) (dual of [3130, 3104, 8]-code), using
- construction X applied to Ce(6) ⊂ Ce(5) [i] based on
- linear OA(526, 3125, F5, 7) (dual of [3125, 3099, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(521, 3125, F5, 6) (dual of [3125, 3104, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(50, 5, F5, 0) (dual of [5, 5, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(50, s, F5, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(6) ⊂ Ce(5) [i] based on
- 4 step Varšamov–Edel lengthening with (ri) = (2, 0, 0, 0) [i] based on linear OA(526, 3130, F5, 7) (dual of [3130, 3104, 8]-code), using
(28−7, 28, 887264)-Net in Base 5 — Upper bound on s
There is no (21, 28, 887265)-net in base 5, because
- 1 times m-reduction [i] would yield (21, 27, 887265)-net in base 5, but
- the generalized Rao bound for nets shows that 5m ≥ 7 450601 645363 198461 > 527 [i]