Remove this bit:
Code:
<td width="5"{if $shipping[ship_num].shippingid eq $cart.shippingid} class="TableHead"{/if}>
<input type="radio" name="shippingid" value="{$shipping[ship_num].shippingid}"{if $shipping[ship_num].shippingid eq $cart.shippingid} checked="checked"{else} onclick="javascript: this.form.submit();"{/if} />
</td>