Best Known (48−12, 48, s)-Nets in Base 5
(48−12, 48, 522)-Net over F5 — Constructive and digital
Digital (36, 48, 522)-net over F5, using
- net defined by OOA [i] based on linear OOA(548, 522, F5, 12, 12) (dual of [(522, 12), 6216, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(548, 3132, F5, 12) (dual of [3132, 3084, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(548, 3133, F5, 12) (dual of [3133, 3085, 13]-code), using
- construction XX applied to Ce(11) ⊂ Ce(10) ⊂ Ce(8) [i] based on
- linear OA(546, 3125, F5, 12) (dual of [3125, 3079, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(541, 3125, F5, 11) (dual of [3125, 3084, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(536, 3125, F5, 9) (dual of [3125, 3089, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [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(11) ⊂ Ce(10) ⊂ Ce(8) [i] based on
- discarding factors / shortening the dual code based on linear OA(548, 3133, F5, 12) (dual of [3133, 3085, 13]-code), using
- OA 6-folding and stacking [i] based on linear OA(548, 3132, F5, 12) (dual of [3132, 3084, 13]-code), using
(48−12, 48, 2177)-Net over F5 — Digital
Digital (36, 48, 2177)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(548, 2177, F5, 12) (dual of [2177, 2129, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(548, 3133, F5, 12) (dual of [3133, 3085, 13]-code), using
- construction XX applied to Ce(11) ⊂ Ce(10) ⊂ Ce(8) [i] based on
- linear OA(546, 3125, F5, 12) (dual of [3125, 3079, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(541, 3125, F5, 11) (dual of [3125, 3084, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(536, 3125, F5, 9) (dual of [3125, 3089, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [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(11) ⊂ Ce(10) ⊂ Ce(8) [i] based on
- discarding factors / shortening the dual code based on linear OA(548, 3133, F5, 12) (dual of [3133, 3085, 13]-code), using
(48−12, 48, 292358)-Net in Base 5 — Upper bound on s
There is no (36, 48, 292359)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 3552 727593 460948 352492 894198 389993 > 548 [i]