Algebraic Techniques in Coding Theory: Entropy Vectors, Frames, and Constrained Coding

The study of codes, classically motivated by the need to communicate information reliably in the presence of error, has found new life in fields as diverse as network communication, distributed storage of data, and even has connections to the design of linear measurements used in compressive sensing. But in all contexts, a code typically involves exploiting the algebraic or geometric structure underlying an application. In this thesis, we examine several problems in coding theory, and try to gain some insight into the algebraic structure behind them.

The first is the study of the entropy region - the space of all possible vectors of joint entropies which can arise from a set of discrete random variables. Understanding this region is essentially the key to optimizing network codes for a given network. To this end, we employ a group-theoretic method of constructing random variables producing so-called "group-characterizable" entropy vectors, which are capable of approximating any point in the entropy region. We show how small groups can be used to produce entropy vectors which violate the Ingleton inequality, a fundamental bound on entropy vectors arising from the random variables involved in linear network codes. We discuss the suitability of these groups to design codes for networks which could potentially outperform linear coding.

The second topic we discuss is the design of frames with low coherence, closely related to finding spherical codes in which the codewords are unit vectors spaced out around the unit sphere so as to minimize the magnitudes of their mutual inner products. We show how to build frames by selecting a cleverly chosen set of representations of a finite group to produce a "group code" as described by Slepian decades ago. We go on to reinterpret our method as selecting a subset of rows of a group Fourier matrix, allowing us to study and bound our frames' coherences using character theory. We discuss the usefulness of our frames in sparse signal recovery using linear measurements.

The final problem we investigate is that of coding with constraints, most recently motivated by the demand for ways to encode large amounts of data using error-correcting codes so that any small loss can be recovered from a small set of surviving data. Most often, this involves using a systematic linear error-correcting code in which each parity symbol is constrained to be a function of some subset of the message symbols. We derive bounds on the minimum distance of such a code based on its constraints, and characterize when these bounds can be achieved using subcodes of Reed-Solomon codes.

Biodeterioration of wooden boats: a major problem facing marine fisheries

In India, the fishing industry alone incurs an annual loss of over 120 million rupees on account of biodeterioration of wooden fishing craft. None of the timber species, currently in demand for boat-building, possesses an natural bioresistance and will be completely destroyed within 6 to 12 months. Preventive measures against biodeterioration range from application of several indigenous formulations to metallic sheathings and pressure impregnation of wood with preservative chemicals. These methods do not provide lasting protection, as each has its own short-comings and inadequacies. The need for long-term research in the field of marine biodeterioration for improving the efficiency of currently known control measures, with emphasis on application of non-polluting biological methods, is stressed.

Scantlings for small wooden fishing vessels

Studies are carried out to find a suitable basis of specifying scantlings for wooden fishing vessels for India, especially for the range 30' to 50' length overall. Equations of the type y=a (where 'y' is scantling in inches, N is cubic numeral in ft³ and 'a', 'b' are constants) are fitted to the scantling tables (applicable to vessels 50’ and above) available from USA, Newfoundland, Denmark, France and Scotland and they are found to represent the regulations accurately. These lines are corrected for standard frame and beam spacing and molded/sided dimensions to bring them on a common basis for comparison and minimum scantling lines for the main structural members are derived. These lines are extended to cover the range 30' to 50' which is generally outside the range of the above regulations.

Comparison of model predictions and speed trial results for a group of 32' wooden trawlers

Speed trial results of six 32' wooden stern, trawlers built to the same design are analyzed to determine the EHP ship Froude number curves in each case and are compared with the model test results of the same design and for the corresponding, displacement, conditions.

Articulated wooden blades for a horizontal-axis wind turbine. Design, construction and testing.

A pair of blades were constructed following a Tapered Chord, Zero Twist pattern after Anderson. The construction uses the Wood Epoxy Saturation Technique, with a solid Beech main spar and leading edge joined together with laminated veneers of beech forming a D-section; the trailing edge is formed from millimetre ply skins, foam filled to resist compressive loads. This construction leads to an extremely light, flexible blade, with the centres of gravity and torsion well forward, giving good stability. Each blade has three built-in strain gauges, alowing flapwise bending to be measured. Stiffness, and natural frequencies, were measured, to input to a numerical computer model to calculate blade deformation during operation, and to determine stability boundaries of the blade. Preliminary aerodynamic performance measurements are presented and close agreement is found with theory.

Cathodic protection of aluminium sheathed wooden fishing boats with ternary aluminium anodes

Results of laboratory and field trials on cathodic protection of aluminium sheathing in fishing boats by ternary aluminium anodes are presented. The high negative potential of 1.06 V with respect to saturated calomel electrode, its appreciably low anodic polarization and high current output are favourable factors for using the ternary aluminium anodes. The low rate of consumption of the anode material under service trials attests its economic viability.

Investigations on wooden floating materials: the loss of buoyancy under continuous immersion in water and the rate of its redemption by drying

A wooden fishing float under immersion in water for long periods is liable to absorb water, the quantity of water absorbed possibly being dependent upon the physical factors like the specific gravity and the inherent property of the material, the time of soaking and the pressure acting on it. Consequently a wooden float is likely to become heavy and loss its original buoyancy. However, when the float is removed from water and dried, the lost buoyancy is regained on complete drying. The present paper is an attempt to elucidate these two important characteristics of some of the chief wooden floating materials used on the West Coast of India.