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If the rm is an imperfect competitor in the commodity market, the MRP declines both because the MPP declines and because the rm must lower the commodity price in order to sell more units. If the rm remains a perfect competitor in the resource market, the rm again maximizes total pro ts when it hires the resource until MRP equals the resource price. The declining MRP schedule then represents the rm s demand schedule for the variable resource. Example 16.1 In Table 16.1, column 1 refers to units of a variable resource, say, labor, employed in a given plant. Column 2 gives the total product produced. Column 3 gives the marginal physical product or the change in total product per unit change in the use of the resource. Commodity price (column 4) declines because of imperfect competition in the commodity market. TR (column 5) is obtained by multiplying commodity price by total product. Column 6 gives the MRP, measured as the change in total revenue. MRP declines both because MPP declines and because the product price declines. A rm that is a perfect competitor in the resource market would maximize its total pro ts by employing the resource until the MRP equals the resource price. Table 16.1

Table 11.3 Some parameters for BCH codes n 7 15 15 15 31 31 31 31 31 k 4 11 7 5 26 21 16 11 6 t 1 1 2 3 1 2 3 5 7

experience errors as a result of impulse-type noise or impulse-type interference Reed-Solomon (R-S) codes are designed to correct errors under these conditions Instead of encoding directly in bits, the bits are grouped into symbols, and the datawords and codewords are encoded in these symbols Errors affecting a group of bits are most likely to affect only one symbol that can be corrected by the R-S code Let the number of bits per symbol be k; then the number of possible k symbols is q 2 (referred to as a q-ary alphabet) Let K be the number of symbols in a dataword and N be the number of symbols in a codeword Just as in the block code where k-bit datawords were mapped into n-bit codewords, with the R-S code, datawords of K symbols are mapped into codewords of N symbols.

The additional N K redundant symbols are derived from the message symbols but are not part of the message The number of possible codewords is 2N, but only 2K of these will contain datawords, and these are the ones that are transmitted It follows that the rest of the codewords are redundant, but only in the sense that they do not contribute to the message If errors occur in transmission, there is high probability that they will convert the permissible codewords into one or another of the redundant words that the decoder at the receiver is designed to recognize as an error It will be noted that the term high probability is used There is always the possibility, however remote, that enough errors occur to transform a transmitted codeword into another legitimate codeword even though this was not the one transmitted.

It will be observed that the wording of the preceding paragraph parallels that given in Sec 112 on block codes, except that here the coding is carried out on symbols Some of the design rules for the R-S codes are q N 2t 2 q N.

CHAPTER 1 THE VBA INTEGRATED DEVELOPMENT ENVIRONMENT (VBAIDE)

Here, t is the number of symbol errors that can be corrected. A simple example will be given to illustrate these. Let k 2; then q 4, and these four symbols may be labeled A, B, C, and D. In terms of the binary symbols (bits) for this simple case, we could have A 00, B 01, C 10, and D 11. One could imagine the binary numbers 00, 01, 10, and 11 being stored in memory locations labeled A, B, C, and D. The number of symbols per codeword is N q 1 3. Suppose that t 1; then the rule 2t N K gives K 1; that is, there will be one symbol per dataword. Hence the number of datawords is qK 4 (i.e., A, B, C, or D), and the number of codewords is qN 43 64. These will include permissible words of the form AP1P2, BP3P4, CP5P6, and DP7P8,

The declining MRP schedule of columns 6 and 1 in Table 16.1 is the rm s demand schedule for the resource and is graphed as d in Figure 16-1. At the resource price of $50, the rm will hire one unit of the resource. At the resource price of $31, the rm will hire two units of the resource, and so on.