535 research outputs found
The MacWilliams Identity for Krawtchouk Association Schemes
The weight distribution of an error correcting code is a crucial statistic in determining its performance. One key tool for relating the weight of a code to that of its dual is the MacWilliams Identity, first developed for the Hamming association scheme. This identity has two forms: one is a functional transformation of the weight enumerators, while the other is a direct relation of the weight distributions via eigenvalues of the association scheme. The functional transformation form can, in particular, be used to derive important moment identities for the weight distribution of codes. In this thesis, we focus initially on extending the functional transformation to codes based on skew-symmetric and Hermitian matrices. A generalised b-algebra and new fundamental homogeneous polynomials are then identified and proven to generate the eigenvalues of a specific subclass of association schemes, Krawtchouk association schemes. Based on the new set of MacWilliams Identities as a functional transform, we derive several moments of the weight distribution for all of these codes
The spectrum of the Neumann Poincaré operator on bow-tie curves
The study of the spectral properties of those operators which output potential functions has historically been of great value, particularly in the resolution of
certain problems in potential theory and the mathematical study of gravitational
and electromagnetic fields. The initial inspiration for this work was to build on
the existing body of results that describe the spectrum of the Neumann Poincar´e
operator on different surfaces, particularly those with corners and edges.
The particular spectral properties of this integral operator align nicely with a
number of scenarios discussed in physics, most notably in the study of plasmonics, where there is a noted coincidence between the elements of the spectrum of
the Neumann Poincar´e operator when acting on specific function spaces and the
phenomena of plasmon resonances.
This work is a study of the spectral properties of the Neumann Poincar´e operator
when considered over sets bounded by bow-tie curves, formed of two tear drop
shapes or ’wings’, each with a corner that coincides, a scenario that distinguishes
itself from prior studies in that the surface being acted on is neither completely
smooth nor can it be characterized as a Lipschitz domain in the region of the
curve’s singular point
LIPIcs, Volume 261, ICALP 2023, Complete Volume
LIPIcs, Volume 261, ICALP 2023, Complete Volum
Monodromy of subrepresentations and irreducibility of low degree automorphic Galois representations
Let be a smooth, separated, geometrically connected scheme defined over a
number field and a system of n-dimensional
semisimple -adic representations of the \'etale fundamental group of
such that for each closed point of , the specialization
is a compatible system of Galois representations
under mild local conditions. For almost all , we prove that any type A
irreducible subrepresentation of is
residually irreducible. When is totally real or CM, , and
is the compatible system of Galois representations
of attached to a regular algebraic, polarized, cuspidal automorphic
representation of , for almost all we
prove that is (i) irreducible and
(ii) residually irreducible if in addition .Comment: Revised version. To appear in JLM
The Shimura lift and congruences for modular forms with the eta multiplier
The Shimura correspondence is a fundamental tool in the study of
half-integral weight modular forms. In this paper, we prove a Shimura-type
correspondence for spaces of half-integral weight cusp forms which transform
with a power of the Dedekind eta multiplier twisted by a Dirichlet character.
We prove that the lift of a cusp form of weight and level has
weight and level , and is new at the primes and with
specified Atkin-Lehner eigenvalues. This precise information leads to
arithmetic applications. For a wide family of spaces of half-integral weight
modular forms we prove the existence of infinitely many primes which
give rise to quadratic congruences modulo arbitrary powers of .Comment: 44 page
Easily decoded error correcting codes
This thesis is concerned with the decoding aspect of linear block error-correcting codes. When, as in most practical situations, the decoder cost is limited an optimum code may be inferior in performance to a longer sub-optimum code' of the same rate. This consideration is a central theme of the thesis.
The best methods available for decoding short optimum codes and long B.C.H. codes are discussed, in some cases new decoding algorithms for the codes are introduced.
Hashim's "Nested" codes are then analysed. The method of nesting codes which was given by Hashim is shown to be optimum - but it is seen that the codes are less easily decoded than was previously thought.
"Conjoined" codes are introduced. It is shown how two codes with identical numbers of information bits may be "conjoined" to give a code with length and minimum distance equal to the sum of the respective parameters of the constituent codes but with the same number of information bits. A very simple decoding algorithm is given for the codes whereby each constituent codeword is decoded and then a decision is made as to the correct decoding. A technique is given for adding more codewords to conjoined codes without unduly increasing the decoder complexity.
Lastly, "Array" codes are described. They are formed by making parity checks over carefully chosen patterns of information bits arranged in a two-dimensional array. Various methods are given for choosing suitable patterns. Some of the resulting codes are self-orthogonal and certain of these have parameters close to the optimum for such codes. A method is given for adding more codewords to array codes, derived from a process of augmentation known for product codes
Quantum modularity for a closed hyperbolic 3-manifold
This paper proves quantum modularity of both functions from and
-series associated to the closed manifold obtained by surgery on the
figure-eight knot, . In a sense, this is a companion to work of
Garoufalidis-Zagier where similar statements were studied in detail for some
simple knots. It is shown that quantum modularity for closed manifolds provides
a unification of Chen-Yang's volume conjecture with Witten's asymptotic
expansion conjecture. Additionally we show that is a
counter-example to previous conjectures relating the Witten-Reshetikhin-Turaev
invariant and the series. This could be reformulated in terms
of a "strange identity", which gives a volume conjecture for the
invariant. Using factorisation of state integrals, we give conjectural but
precise -hypergeometric formulae for generating series of Stokes constants
of this manifold. We find that the generating series of Stokes constants is
related to the 3d index of Dimofte-Gaiotto-Gukov of proposed by
Gang-Yonekura. This extends the equivalent conjecture of
Garoufalidis-Gu-Mari\~no for knots to closed manifolds. This work appeared in a
similar form in the author's thesis.Comment: 73 pages, 6 figures, (v2 added a conjectural link between the 3d
index and generating series of Stokes constants
Faster Complete Formulas for the GLS254 Binary Curve
GLS254 is an elliptic curve defined over a finite field of characteristic 2; it contains a 253-bit prime order subgroup, and supports an endomorphism that can be efficiently computed and helps speed up some typical operations such as multiplication of a curve element by a scalar. That curve offers on x86 and ARMv8 platforms the best known performance for elliptic curves at the 128-bit security level.
In this paper we present a number of new results related to GLS254:
- We describe new efficient and complete point doubling formulas (2M+4S) applicable to all ordinary binary curves.
- We apply the previously described (x,s) coordinates to GLS254, enhanced with the new doubling formulas. We obtain formulas that are not only fast, but also complete, and thus allow generic constant-time usage in arbitrary cryptographic protocols.
- Our strictly constant-time implementation multiplies a point by a scalar in 31615 cycles on an x86 Coffee Lake, and 77435 cycles on an ARM Cortex-A55, improving previous records by 13% and 11.7% on these two platforms, respectively.
- We take advantage of the completeness of the formulas to define some extra operations, such as canonical encoding with (x, s) compression, constant-time hash-to-curve, and signatures. Our Schnorr signatures have size only 48 bytes, and offer good performance: signature generation in 18374 cycles, and verification in 27376 cycles, on x86; this is about four times faster than the best reported Ed25519 implementations on the same platform.
- The very fast implementations leverage the carryless multiplication opcodes offered by the target platforms. We also investigate performance on CPUs that do not offer such an operation, namely a 64-bit RISC-V CPU (SiFive-U74 core) and a 32-bit ARM Cortex-M4 microcontroller. While the achieved performance is substantially poorer, it is not catastrophic; on both platforms, GLS254 signatures are only about 2x to 2.5x slower than Ed25519
The Fifteenth Marcel Grossmann Meeting
The three volumes of the proceedings of MG15 give a broad view of all aspects of gravitational physics and astrophysics, from mathematical issues to recent observations and experiments. The scientific program of the meeting included 40 morning plenary talks over 6 days, 5 evening popular talks and nearly 100 parallel sessions on 71 topics spread over 4 afternoons. These proceedings are a representative sample of the very many oral and poster presentations made at the meeting.Part A contains plenary and review articles and the contributions from some parallel sessions, while Parts B and C consist of those from the remaining parallel sessions. The contents range from the mathematical foundations of classical and quantum gravitational theories including recent developments in string theory, to precision tests of general relativity including progress towards the detection of gravitational waves, and from supernova cosmology to relativistic astrophysics, including topics such as gamma ray bursts, black hole physics both in our galaxy and in active galactic nuclei in other galaxies, and neutron star, pulsar and white dwarf astrophysics. Parallel sessions touch on dark matter, neutrinos, X-ray sources, astrophysical black holes, neutron stars, white dwarfs, binary systems, radiative transfer, accretion disks, quasars, gamma ray bursts, supernovas, alternative gravitational theories, perturbations of collapsed objects, analog models, black hole thermodynamics, numerical relativity, gravitational lensing, large scale structure, observational cosmology, early universe models and cosmic microwave background anisotropies, inhomogeneous cosmology, inflation, global structure, singularities, chaos, Einstein-Maxwell systems, wormholes, exact solutions of Einstein's equations, gravitational waves, gravitational wave detectors and data analysis, precision gravitational measurements, quantum gravity and loop quantum gravity, quantum cosmology, strings and branes, self-gravitating systems, gamma ray astronomy, cosmic rays and the history of general relativity
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