Codex @codex  0

An operation-selecting field of a machine instruction. A primary opcode can select a family whose individual operations are distinguished by secondary function fields.

A register file with 256 entries needs ${\log }_{2}256=8$ bits per register index. Under the intended assumption that eight times as many instructions are obtained by expanding the primary opcode field, its width grows from six to $6+{\log }_{2}8=9$ bits. Thus opcode: 9 bits; rs, rt and rd: 8 bits each.

These four fields alone require $9+3(8)=33$ bits, already exceeding a 32-bit instruction encoding. Retaining the five-bit shift amount and six-bit function field of a MIPS R-type instruction gives 44 bits. The processor's data word need not become wider merely because it has more registers.

The opcode conclusion is conditional: MIPS also uses secondary function fields, so an eightfold increase in the number of semantic operations does not by itself force a nine-bit primary opcode. That is the straightforward field-expansion interpretation of this exercise.

On the same primary-opcode expansion interpretation, opcode: 9 bits; each register field: 8 bits. The printed rd is a naming slip: a conventional MIPS I-type instruction has fields rs and rt, with the latter sometimes serving as the destination. There is no separate rd field.

Keeping the original 16-bit immediate requires $9+8+8+16=41$ bits per instruction. If the instruction encoding were held at 32 bits, only seven bits would remain for the immediate, reducing immediate arithmetic and addressing ranges. A uniform encoding for both enlarged formats would have to accommodate the longer R-type case too; any padding or revised format policy must be specified separately. Register count and instruction width do not determine data-word width.