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The structure of free resolutions of finite length modules over regular local rings has long been a topic of interest in commutative algebra. Conjectures by Buchsbaum-Eisenbud-Horrocks and Avramov-Buchweitz predict that in this setting the minimal free resolution of the residue field should give, in some sense, the smallest possible free resolution of a finite length module. Results of Tate and Shamash describing the minimal free resolution of the residue field over a local hypersurface ring, together with the theory of matrix factorizations developed by Eisenbud and Eisenbud-Peeva, suggest analogous lower bounds for the size of free resolutions of finite length modules of infinite projective dimension over such rings. In this dissertation we describe both positive and negative results pertaining to these lower bounds. By refining an argument of Charalambous, we show that the lower bounds hold in certain multigraded settings. We are also able to obtain results for finite free resolutions of multigraded modules, recovering results of Charalambous and Santoni. For the local case, however, we use a construction of Iyengar-Walker to provide examples showing that the lower bounds do not always hold. In order to accomplish this, we make use of the theory of higher matrix factorizations developed by Eisenbud-Peeva to investigate the structure of free resolutions over complete intersections of arbitrary codimension.

Magnetic field and eddy currents in a cylinder of finite length are calculated by separation of variables. The magnetic field outside the cylinder or inside the bore of the hollow cylinder and shell is expressed in terms of Bessel functions. Both axial and transverse applied fields are considered for the solid and hollow cylinders. The equations for the vector potential components are transformed in one-dimensional equations along the radial coordinate with the consequent integration   by the method of variation of parameters. The equation for the scalar electric potential when required is also integrated analytically. Expressions for the magnetic moment and loss are derived. An alternative analytical solution in terms of scalar magnetic potential is derived for the finite length thin shells. All formulas are validated by the comparison with the solutions by finite–element and finite-difference methods.

In this paper, we propose a methodology to compute the optimal finite-length coding rate for random linear network coding schemes over a line network. To do so, we first model the encoding, reencoding, and decoding process of different coding schemes in matrix notation and corresponding error probabilities. Specifically, we model the finite-length performance for random linear capacity-achieving schemes: non-systematic (RLNC) and systematic (SNC) and non-capacity achieving schemes: SNC with packet scheduling (SNC-S) or sliding window (SWNC). Then, we propose a binary searching algorithm to identify optimal coding rate for given target packet loss rate. We use our proposed method to obtain the region of exponential increase of optimal coding rate and corresponding slopes for representative types of traffic and erasure rates. Our results show the tradeoff for capacity-achieving codes vs non-capacity achieving schemes, since the latter trade throughput with delay, which is reflected in the decrease of the exponential slope with the blocklength. We also show the effect of the number of re-encoding times, which further decreases the slope., Comment: 11 pages, 5 figures

Secure communications is becoming increasingly relevant in the development of space technology. Well established cryptographic technology is already in place and is expected to continue to be so. On the other hand, information theoretical security emerges as a post-quantum versatile candidate to complement overall security strength. In order to prove such potential, performance analysis methods are needed that consider realistic legitimate and eavesdropper system assumptions and non-asymptotic coding lengths. In this paper we propose the design of secure radio frequency (RF) satellite links with realistic system assumptions. Our contribution is three-fold. First, we propose a wiretap channel model for the finite-length regime. The model includes an stochastic wiretap encoding method using existing practical linear error correcting codes and hash codes. Secrecy is provided with privacy amplification, for which the finite-length secrecy metric is given that upper bounds semantic secrecy. Second, we derive a novel RF (broadcast) satellite wiretap channel model that parameterizes the stochastic degraded channel around the legitimate channel, a necessary condition to enable secure communication. Finally, we show the design of a secure satellite physical layer and finite-length performance evaluation. In doing so, we define as sacrifice rate the fixed fraction of the overall coding rate budget for reliability that needs to be allocated to secrecy. Our methodology does not make use of channel side information of the eavesdropper, only assumes worst case system assumptions. We illustrate our proposed design method with numerical results using practical error correcting codes in current standards of satellite communication., Comment: Submitted to IEEE journal Corrected typo in eq. (18) and its derivation eq. (46). arXiv admin note: text overlap with arXiv:1610.07252

In this paper, we investigate the optimum way to allocate redundancy of finite-length nested codes for modern nonvolatile memories suffering from both permanent defects and transient errors (erasures or random errors). A nested coding approach such as partitioned codes can handle both permanent defects and transient errors by using two parts of redundancy: 1) redundancy to deal with permanent defects and 2) redundancy for transient errors. We consider two different channel models of the binary defect and erasure channel (BDEC) and the binary defect and symmetric channel (BDSC). The transient errors of the BDEC are erasures and the BDSC's transient errors are modeled by the binary symmetric channel, respectively. Asymptotically, the probability of recovery failure can converge to zero if the capacity region conditions of nested codes are satisfied. However, the probability of recovery failure of finite-length nested codes can be significantly variable for different redundancy allocations even though they all satisfy the capacity region conditions. Hence, we formulate the redundancy allocation problem of finite-length nested codes to minimize the recovery failure probability. We derive the upper bounds on the probability of recovery failure and use them to estimate the optimal redundancy allocation. Numerical results show that our estimated redundancy allocation matches well the optimal redundancy allocation., Comment: accepted by Journal of Communications and Networks (JCN)