An explicit representation and enumeration for negacyclic codes of length 2^kn over \mathbb{Z}_4+u\mathbb{Z}_4

Abstract

In this paper, we give an explicit representation and enumeration for negacyclic codes of length 2 k n over the local non-principal ideal ring R = Z 4 + uZ 4 (u 2 = 0), where k, n are arbitrary positive integers and n is odd. In particular, we present all distinct negacyclic codes of length 2 k over R precisely. Moreover, we provide an exact mass formula for the number of negacyclic codes of length 2 k n over R and correct several mistakes in some literatures.

The Hunt for Pevatrons: The Case of Supernova Remnants

“…A prime example of a gamma-ray emission interpreted as the result of accelerated electrons is the Crab Nebula, the brightest source of the TeV sky, and thus one of the best studied objects [26][27][28]. More generally, several theoretical works have illustrated that electrons accelerated above 100 TeV could lead, through inverse Compton scattering or bremsstrahlung, to gamma rays in the 100 TeV range (see e.g., [9] and references therein), and can produce hard gamma-ray spectra extending up to the 100 TeV, making it especially strenuous to differentiate between hadronic and leptonic acceleration mechanisms [29].…”

“…was capable of claiming the detection of the first pevatron in the Galactic center region [39]. Other observatories, using different techniques, relying on shower front detectors (or "air shower detectors"), such as HAWC [40], Tibet ASγ [41] and LHAASO [28], operating up to and above the ∼100 TeV gamma-ray domain also reported on the detection of several Galactic pevatrons. All these observations indicated that most of the pevatron candidates seem to not be associated to SNRs.…”

“…Discussions on the acceleration mechanisms usually invoke magnetic reconnection around the Crab pulsar and shock acceleration at the pulsar-wind termination shocks [108][109][110][111][112][113][114][115]. Recently, the LHAASO instrument reported on the detection of gamma rays above 1 PeV, additional confirmation that an electron pevatron is located within the Crab Nebulae [28,116]. This means that, unexpectedly, the first detection of a pevatron corresponded to an electronpevatron, a reminder that the finding of a pevatron does not immediately solve all problems of CR physics.…”

“…Finally, with the detection of gamma rays of energy 100 TeV by LHAASO from a dozen of sources that are likely located in the Galaxy, another important milestone has been reached: the first detection of a population of pevatrons [28]. So far, the spectral shape of most of these sources is still not well established, which makes it difficult to clearly understand whether the origin is leptonic or hadronic.…”

“…First, stricto census, current IACTs are not sensitive enough to detect gamma rays in ∼100 TeV range, but they still have been able to measure the gamma-ray spectrum in the TeV and tens of TeV region, and revealing exponential suppression that indicate that the flux in the 100 TeV range should be at least drastically suppressed. Second, recent results of several instruments, HAWC, TibetASγ and LHAASO [28,68,69] have reported on the detection of ∼100 TeV gamma rays from a region in which the SNR G106.3+2.7 is located. By design, the improved sensitivity in the 100 TeV range of these observatories, compared to typical IACTs, comes with a poorer angular resolution than typical IACTs.…”