Consider a processor with 16 general-purpose registers and a 2-byte…

Concept

For a fixed 16-bit instruction word with variable-length prefix opcodes, each opcode reserves every bit pattern formed by its operand fields. Therefore, the fractions of the instruction space occupied by all instruction classes must add to at most 1.

If a class has k operand bits and n opcodes, it occupies n × 2k encodings out of 216. This is the prefix-space constraint expressed by the Kraft inequality.

Application

1. Each register number needs log2(16) = 4 bits.

2. An M-type instruction has 2 × 4 + 6 = 14 operand bits. Its 2 opcodes occupy 2 × 214 = 32768 encodings.

3. An R-type instruction has 3 × 4 = 12 operand bits. Its 7 opcodes occupy 7 × 212 = 28672 encodings.

4. A C-type instruction has 4 + 6 = 10 operand bits. If N C-type opcodes are provided, they occupy N × 210 encodings.

5. The total cannot exceed 216: 2 × 214 + 7 × 212 + N × 210 ≤ 216.

6. Dividing by 210 gives 32 + 28 + N ≤ 64, so N ≤ 4. Therefore, the maximum number of C-type opcodes is 4.

Cross-check

Using opcode-prefix fractions directly: 2/22 + 7/24 + N/26 ≤ 1. Thus 1/2 + 7/16 + N/64 ≤ 1, which again gives N ≤ 4.