Efficient Multi-Point Local Decoding of Reed-Muller Codes via Interleaved Codex

Reed-Muller codes are among the most important classes of locally correctable codes. Currently local decoding of Reed-Muller codes is based on decoding on lines or quadratic curves to recover one single coordinate. To recover multiple coordinates simultaneously, the naive way is to repeat the local decoding for recovery of a single coordinate. This decoding algorithm might be more expensive, i.e., require higher query complexity. In this paper, we focus on Reed-Muller codes with usual parameter regime, namely, the total degree of evaluation polynomials is $d=\Theta({q})$, where $q$ is the code alphabet size (in fact, $d$ can be as big as $q/4$ in our setting). By introducing a novel variation of codex, i.e., interleaved codex (the concept of codex has been used for arithmetic secret sharing \cite{C11,CCX12}), we are able to locally recover arbitrarily large number $k$ of coordinates of a Reed-Muller code simultaneously at the cost of querying $O(q^2k)$ coordinates. It turns out that our local decoding of Reed-Muller codes shows ({\it perhaps surprisingly}) that accessing $k$ locations is in fact cheaper than repeating the procedure for accessing a single location for $k$ times. Our estimation of success error probability is based on error probability bound for $t$-wise linearly independent variables given in \cite{BR94}

Early texts on Hindu-Arabic calculation

This article describes how the decimal place value system was transmitted from India via the Arabs to the West up to the end of the fifteenth century. The arithmetical work of al-Khw¯arizm¯ı’s, ca. 825, is the oldest Arabic work on Indian arithmetic of which we have detailed knowledge. There is no known Arabic manuscript of this work; our knowledge of it is based on an early reworking of a Latin translation. Until some years ago, only one fragmentary manuscript of this twelfth-century reworking was known (Cambridge, UL, Ii.6.5). Another manuscript that transmits the complete text (New York, Hispanic Society of America, HC 397/726) has made possible a more exact study of al-Khw¯arizm¯ı’s work. This article gives an outline of this manuscript’s contents and discusses some characteristics of its presentation

The Mayan Long Count Calendar

The Maya were known for their astronomical proficiency. This is demonstrated in the Mayan codices where ritual practices were related to astronomical events/predictions. Whereas Mayan mathematics were based on a vigesimal system, they used a different base when dealing with long periods of time, the Long Count Calendar (LCC), composed of different Long Count Periods: the Tun of 360 days, the Katun of 7200 days and the Baktun of 144000 days. There were two other calendars used in addition to the LCC: a civil year Haab' of 365 days and a religious year Tzolk'in of 260 days. Another cycle was also in use: a 3276-day cycle (combination of the 819-day Kawil cycle and the 4 directions-colors). Based on the hypothesis that Mayan astronomers had a good knowledge of naked-eye astronomy, we propose here an explanation of the origin of the LCC, the Haab', the Tzolk'in and the 3276-day cycle. This study sheds more light on the connection between arithmetic, astronomy and religion in Maya civilization

Eyewitness to Jesus: amazing new manuscript evidence about the origin of the Gospels

Reviewed Book: Thiede, Carsten Peter. Eyewitness to Jesus: amazing new manuscript evidence about the origin of the Gospels. New York: Doubleday, 1996

Efficient multi-point local decoding of Reed-Muller codes via interleaved codex

Reed-Muller codes are among the most important classes of locally correctable codes. Currently local decoding of Reed-Muller codes is based on decoding on lines or quadratic curves to recover one single coordinate. To recover multiple coordinates simultaneously, the naive way is to repeat the local decoding for recovery of a single coordinate. This decoding algorithm might be more expensive, i.e., require higher query complexity. In this paper, we focus on Reed-Muller codes with usual parameter regime, namely, the total degree of evaluation polynomials is d=Θ {q), where q is the code alphabet size (in fact, d can be as big as q/4 in our setting). By introducing a novel variation of codex, i.e., interleaved codex (the concept of codex has been used for arithmetic secret sharing), we are able to locally recover arbitrarily large number k of coordinates of a Reed-Muller code simultaneously with error probability exp (-Ω (k)) at the cost of querying merely O(q2k) coordinates. It turns out that our local decoding of Reed-Muller codes shows (perhaps surprisingly) that accessing k locations is in fact cheaper than repeating the procedure for accessing a single location for k times. Precisely speaking, to get the same success probability by repeating the local decoding algorithm of a single coordinate, one has to query Ω (qk2) coordinates. Thus, the query complexity of our local decoding is smaller for k=Ω (q). If we impose the same query complexity constraint on both algorithm, our local decoding algorithm yields smaller error probability when k=Ω (qq). In addition, our local decoding is efficient, i.e., the decoding complexity is Poly(k,q). Construction of an interleaved codex is based on concatenation of a codex with a multiplication friendly pair, while the main tool to realize codex is based on algebraic function fields (or more precisely, algebraic geometry codes)

Waiting for the barbarians : seeking solutions or awaiting answers?

Cavafy's (1961) famous poem describes the paralysis induced when waiting for an event that never happens, finding distraction in other things while expecting imminent cataclysm. Libraries have perhaps been guilty of this in recent years, joining the chorus bemoaning the imminent arrival of the digital barbarians and undertaking a whole series of avoidance tactics which make libraries ever more efficient but ever less relevant. Like Cavafy's senators, libraries have prepared themselves, decked themselves in finery but have waited for the future to come to them rather than gone out to engage with it

Безопасность и качество пищевых продуктов = Practical Food Safety and Food Quality : практикум

Даны описания практических и лабораторных работ, проводимых в рамках курсов «Международные стандарты и безопасность продуктов питания» и «Контроль качества биотехнологических продуктов». Задания сопровождаются подробными комментариями. Используются активные формы обучения, такие как работа в команде, игровые технологии и пр. Для иностранных и российских студентов, обучающихся на английском языке, изучающих пищевую биотехнологию, контроль качества и безопасность пищевых продуктов

The Birth of Pictoriality in Computer Media

The aim of the paper is to follow some milestones of the story of computer media as far as the notion of pictoriality is concerned. I am going to describe in the most general way how it happens that two quite separate technologies as computer machine and pictorial representation met and since then became almost inseparable