MinT - Best Known (99, 122, s)-Nets in Base 4

Best Known (99, 122, s)-Nets in Base 4

(99, 122, 1490)-Net over F₄ — Constructive and digital

Digital (99, 122, 1490)-net over F₄, using

4² times duplication [i] based on digital (97, 120, 1490)-net over F₄, using
- net defined by OOA [i] based on linear OOA(4¹²⁰, 1490, F₄, 23, 23) (dual of [(1490, 23), 34150, 24]-NRT-code), using
  - OOA 11-folding and stacking with additional row [i] based on linear OA(4¹²⁰, 16391, F₄, 23) (dual of [16391, 16271, 24]-code), using
    - constructionÂ X applied to C_e(22) âŠ‚ C_e(21) [i] based on
      1. linear OA(4¹²⁰, 16384, F₄, 23) (dual of [16384, 16264, 24]-code), using an extension C_e(22) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 4⁷âˆ’1, defining interval IÂ =Â [1,22], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 23 [i]
      2. linear OA(4¹¹³, 16384, F₄, 22) (dual of [16384, 16271, 23]-code), using an extension C_e(21) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 4⁷âˆ’1, defining interval IÂ =Â [1,21], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 22 [i]
      3. 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
        OA with only one run / dual of code without redundancy [i]

(99, 122, 8502)-Net over F₄ — Digital

Digital (99, 122, 8502)-net over F₄, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(4¹²², 8502, F₄, 23) (dual of [8502, 8380, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(4¹²², 16400, F₄, 23) (dual of [16400, 16278, 24]-code), using
  - 1 times code embedding in larger space [i] based on linear OA(4¹²¹, 16399, F₄, 23) (dual of [16399, 16278, 24]-code), using
    - constructionÂ X applied to C_e(22) âŠ‚ C_e(20) [i] based on
      1. linear OA(4¹²⁰, 16384, F₄, 23) (dual of [16384, 16264, 24]-code), using an extension C_e(22) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 4⁷âˆ’1, defining interval IÂ =Â [1,22], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 23 [i]
      2. linear OA(4¹⁰⁶, 16384, F₄, 21) (dual of [16384, 16278, 22]-code), using an extension C_e(20) of the primitive narrow-sense BCH-code C(I) with length 16383Â = 4⁷âˆ’1, defining interval IÂ =Â [1,20], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 21 [i]
      3. linear OA(4¹, 15, F₄, 1) (dual of [15, 14, 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
        parity-check code [i]

(99, 122, 6863604)-Net in Base 4 — Upper bound on s

There is no (99, 122, 6863605)-net in base 4, because

1 times m-reduction [i] would yield (99, 121, 6863605)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 7 067392 600295 615365 214329 179450 333870 842587 862849 432950 262022 388818 649096 > 4¹²¹ [i]