8,821 research outputs found

    Further results on covering codes with radius R and codimension tR + 1

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    The length function β„“q(r,R)\ell_q(r,R) is the smallest possible length nn of a q q -ary linear [n,nβˆ’r]qR[n,n-r]_qR code with codimension (redundancy) rr and covering radius RR. Let sq(N,ρ)s_q(N,\rho) be the smallest size of a ρ\rho-saturating set in the projective space PG(N,q)\mathrm{PG}(N,q). There is a one-to-one correspondence between [n,nβˆ’r]qR[n,n-r]_qR codes and (Rβˆ’1)(R-1)-saturating nn-sets in PG(rβˆ’1,q)\mathrm{PG}(r-1,q) that implies β„“q(r,R)=sq(rβˆ’1,Rβˆ’1)\ell_q(r,R)=s_q(r-1,R-1). In this work, for Rβ‰₯3R\ge3, new asymptotic upper bounds on β„“q(tR+1,R)\ell_q(tR+1,R) are obtained in the following form: βˆ™Β β„“q(tR+1,R)=sq(tR,Rβˆ’1)≀R!RRβˆ’2Rβ‹…q(rβˆ’R)/Rβ‹…ln⁑qR+o(q(rβˆ’R)/R),r=tR+1,Β tβ‰₯1,Β qΒ isΒ anΒ arbitraryΒ primeΒ power,Β qΒ isΒ largeΒ enough;\hspace{0.7cm} \bullet~\ell_q(tR+1,R) =s_q(tR,R-1)\le \sqrt[R]{\frac{R!}{R^{R-2}}}\cdot q^{(r-R)/R}\cdot\sqrt[R]{\ln q}+o(q^{(r-R)/R}), \hspace{0.3cm}r=tR+1,~t\ge1,~ q\text{ is an arbitrary prime power},~q\text{ is large enough}; βˆ™Β Β ifΒ additionallyΒ RΒ isΒ largeΒ enough,Β thenΒ R!RRβˆ’2R∼1eβ‰ˆ0.3679.\hspace{0.7cm} \bullet~\text{ if additionally }R\text{ is large enough, then }\sqrt[R]{\frac{R!}{R^{R-2}}}\thicksim\frac{1}{e}\thickapprox0.3679. The new bounds are essentially better than the known ones. For t=1t=1, a new construction of (Rβˆ’1)(R-1)-saturating sets in the projective space PG(R,q)\mathrm{PG}(R,q), providing sets of small sizes, is proposed. The [n,nβˆ’(R+1)]qR[n,n-(R+1)]_qR codes, obtained by the construction, have minimum distance R+1R + 1, i.e. they are almost MDS (AMDS) codes. These codes are taken as the starting ones in the lift-constructions (so-called "qmq^m-concatenating constructions") for covering codes to obtain infinite families of codes with growing codimension r=tR+1r=tR+1, tβ‰₯1t\ge1.Comment: 24 pages. arXiv admin note: text overlap with arXiv:2108.1360

    On upper bounds on the smallest size of a saturating set in a projective plane

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    In a projective plane Ξ q\Pi _{q} (not necessarily Desarguesian) of order q,q, a point subset SS is saturating (or dense) if any point of Ξ qβˆ–S\Pi _{q}\setminus S is collinear with two points inΒ S~S. Using probabilistic methods, the following upper bound on the smallest size s(2,q) s(2,q) of a saturating set in Ξ q\Pi _{q} is proved: \begin{equation*} s(2,q)\leq 2\sqrt{(q+1)\ln (q+1)}+2\thicksim 2\sqrt{q\ln q}. \end{equation*} We also show that for any constant cβ‰₯1c\ge 1 a random point set of size kk in Ξ q\Pi _{q} with 2c(q+1)ln⁑(q+1)+2≀k<q2βˆ’1q+2∼q 2c\sqrt{(q+1)\ln(q+1)}+2\le k<\frac{q^{2}-1}{q+2}\thicksim q is a saturating set with probability greater than 1βˆ’1/(q+1)2c2βˆ’2.1-1/(q+1)^{2c^{2}-2}. Our probabilistic approach is also applied to multiple saturating sets. A point set SβŠ‚Ξ qS\subset \Pi_{q} is (1,ΞΌ)(1,\mu)-saturating if for every point QQ of Ξ qβˆ–S\Pi _{q}\setminus S the number of secants of SS through QQ is at least ΞΌ\mu , counted with multiplicity. The multiplicity of a secant β„“ \ell is computed as (#(β„“β€‰βˆ©S)2).{\binom{{\#(\ell \,\cap S)}}{{2}}}. The following upper bound on the smallest size sΞΌ(2,q)s_{\mu }(2,q) of a (1,ΞΌ)(1,\mu)-saturating set in Ξ q\Pi_{q} is proved: \begin{equation*} s_{\mu }(2,q)\leq 2(\mu +1)\sqrt{(q+1)\ln (q+1)}+2\thicksim 2(\mu +1)\sqrt{ q\ln q}\,\text{ for }\,2\leq \mu \leq \sqrt{q}. \end{equation*} By using inductive constructions, upper bounds on the smallest size of a saturating set (as well as on a (1,ΞΌ)(1,\mu)-saturating set) in the projective space PG(N,q)PG(N,q) are obtained. All the results are also stated in terms of linear covering codes.Comment: 15 pages, 24 references, misprints are corrected, Sections 3-5 and some references are adde

    A single-photon sampling architecture for solid-state imaging

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    Advances in solid-state technology have enabled the development of silicon photomultiplier sensor arrays capable of sensing individual photons. Combined with high-frequency time-to-digital converters (TDCs), this technology opens up the prospect of sensors capable of recording with high accuracy both the time and location of each detected photon. Such a capability could lead to significant improvements in imaging accuracy, especially for applications operating with low photon fluxes such as LiDAR and positron emission tomography. The demands placed on on-chip readout circuitry imposes stringent trade-offs between fill factor and spatio-temporal resolution, causing many contemporary designs to severely underutilize the technology's full potential. Concentrating on the low photon flux setting, this paper leverages results from group testing and proposes an architecture for a highly efficient readout of pixels using only a small number of TDCs, thereby also reducing both cost and power consumption. The design relies on a multiplexing technique based on binary interconnection matrices. We provide optimized instances of these matrices for various sensor parameters and give explicit upper and lower bounds on the number of TDCs required to uniquely decode a given maximum number of simultaneous photon arrivals. To illustrate the strength of the proposed architecture, we note a typical digitization result of a 120x120 photodiode sensor on a 30um x 30um pitch with a 40ps time resolution and an estimated fill factor of approximately 70%, using only 161 TDCs. The design guarantees registration and unique recovery of up to 4 simultaneous photon arrivals using a fast decoding algorithm. In a series of realistic simulations of scintillation events in clinical positron emission tomography the design was able to recover the spatio-temporal location of 98.6% of all photons that caused pixel firings.Comment: 24 pages, 3 figures, 5 table
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